how to prove ~(P <--> Q) |- ~P <-->Q
translogi
How do you prove:
~(P <--> Q) |- ~P <--> Q
using natural deduction?
I knew there were some logic provers on the web but lost the links to
them.
can anybody suply them.
Also a more technical question:
I read references to an algorithm for solving natural deduction for
propositional deductions like this one, were can I find the complete
algorithm?
G. Frege
wrote:
>
> How do you prove:
>
> ~(P <-> Q) |- ~P <-> Q
>
> using natural deduction?
>
Proof in Allen and Hand's system (of natural deduction):
1 (1) ~(P <-> Q) A
2 (2) ~P A
2 (3) P -> Q 2 FA
4 (4) ~Q A
4 (5) Q -> P 4 FA
2,4 (6) P <-> Q 3,5 <->I
1,2 (7) Q 1,6 RAA (4)
1 (8) ~P -> Q 7 ->I (2)
9 (9) Q A
9 (10) P -> Q 9 TC
11 (11) P A
11 (12) Q -> P 11 TC
9,11 (13) P <-> Q 10,12 <->I
1,9 (14) ~P 1,13 RAA (11)
1 (15) Q -> ~P 14 ->I (9)
1 (16) ~P <-> Q 8,15 <->I
F.
--
E-mail: info<at>simple-line<dot>de
G. Frege
>>
>> How do you prove:
>>
>> ~(P <-> Q) |- ~P <-> Q
>>
>> using natural deduction?
>>
>
> Proof in Allen and Hand's system (of natural deduction):
>
1 (1) ~(P <-> Q) A
2 (2) ~P A
3 (3) ~Q A
2 (4) P -> Q 2 FA
3 (5) Q -> P 3 FA
2,3 (6) P <-> Q 4,5 <->I
1,2 (7) Q 1,6 RAA (3)
1 (8) ~P -> Q 7 ->I (2)
9 (9) Q A
10 (10) P A
9 (11) P -> Q 9 TC
10 (12) Q -> P 10 TC
9,10 (13) P <-> Q 11,12 <->I
1,9 (14) ~P 1,13 RAA (10)
Charlie-Boo
> I cannot get my head around this one
That's what everyone tells me. :)
> How do you prove:
> ~(P <--> Q) |- ~P <--> Q
> using natural deduction?
1. Create a list of FROM (wff,number)s and TO (wff,numbers)
initialized to (PREMISE,0) and (CONCLUSION,0) respectively.
2. For MAXSIZE=1
Charlie-Boo
1. Keep 2 lists of FROM and TO (wff,number) initialized to (PREMISE,0)
and (CONCLUSION,0) respectively.
2. For MAXSIZE = 1 by 1
3. For all wffs W of size up to MAXSIZE
4. For all FROM pairs (V,N) Do 5
5. For all tuples of Boolean values for the variables Do 6
6. If in every case that V is true wff W is true, add (W,N+1) to the
FROM pairs. End 5,6
7. For all TO pairs (V,N) Do 8
8. For all tuples of Boolean values for the variables Do 9
9. If in every case that W is true wff V is true, add (V,N+1) to the
TO pairs. End 8,9
10. Any WFF that is in a FROM with number M and also a TO with number
N, we can prove PREMISE=>CONCLUSION in M+N+1 steps. Start at PREMISE
and each next step S (1, 2, . . . ) is any wff that is in a FROM with
S and a TO with M+N-S where the S wff is always true when the S-1 wff
is true, by considering all tuples of Boolean values.
11. Stop when you get to the conclusion wff where the number is 0 in
the TO pairs.
Something like this. I just made this up. But it's a general
algorithm where you have to get from A to B going in the minimum
number of steps and can tell which steps you can go to next from any
point. You create a table of the number of steps to go from A to each
wff and from each wff to B.
This is what I've been trying to get Frege to understand.
Propositional Calculus rules of inference are just any pair of wffs
where the second one is always true when the first one is, and you
have to walk through from the premise to the conclusion. So it's a
waste of time to talk about specific rules. You just search for a
path. Rather than having a bunch of rules, you have one algorithm for
determining the distance each wff is from the premise and to the
conclusion.
C-B
G. Frege
wrote:
>
> Propositional Calculus rules of inference are just any pair of wffs
> where the second one is always true when the first one is, and you
> have to walk through from the premise to the conclusion.
>
I'd call that "using a truth-table". :-)
P Q | ~(P <-> Q) | (~P <-> Q)
----+------------|-----------
T T | F T T T | FT F T
T F | T T F F | FT T F <
F T | T F F T | TF T T <
F F | F F T F | TF F F
F.
P.S.
Right. Proofs are superfluous in PC.
On the other hand you might try to show
A -> (B -> (C -> (D -> ... -> (X -> (Y -> Z))...)))
|= ((...(((A & B) & C) & D) & ... & X) & Y) -> Z.
using a truth-table. ( Do it! :-)
R. Srinivasan
> On 6 Feb 2007 21:19:16 -0800, "Charlie-Boo" <shymath...@gmail.com>
> wrote:
>
>
>
> > Propositional Calculus rules of inference are just any pair of wffs
> > where the second one is always true when the first one is, and you
> > have to walk through from the premise to the conclusion.
>
> I'd call that "using a truth-table". :-)
>
> P Q | ~(P <-> Q) | (~P <-> Q)
> ----+------------|-----------
> T T | F T T T | FT F T
> T F | T T F F | FT T F <
> F T | T F F T | TF T T <
> F F | F F T F | TF F F
>
> F.
>
> P.S.
> Right. Proofs are superfluous in PC.
>
> On the other hand you might try to show
>
> A -> (B -> (C -> (D -> ... -> (X -> (Y -> Z))...)))
> |= ((...(((A & B) & C) & D) & ... & X) & Y) -> Z.
>
> using a truth-table. ( Do it! :-)
>
Charlie-Boo, a.k.a Charlie-Bew.
C'mon, Charlie. Sock it to him.
Regards, RS
G. Frege
>
> P.S.
> Right. Proofs are superfluous in PC [in principle].
>
Of course this is rather old news... :-) [See. 6.1262 below.]
----------------------------
6.1 The propositions of logic are tautologies.
6.113 It is the characteristic mark of logical propositions that
one can perceive in the symbol alone that they are true [i.e.
"logically true" --G.F.]
6.126 Whether a proposition belongs to logic can be calculated by
calculating the logical properties of the _symbol_.
And this we do when we prove a logical proposition. For without
troubling ourselves about a sense and a meaning, we form the
logical propositions out of others by mere _symbolic rules_.
We prove a logical proposition by creating it out of other logical
propositions by applying in succession certain operations, which
again generate tautologies out of the first. [...]
Naturally this way of showing that its propositions are tautologies
is quite unessential to logic. Because the propositions, from which
the proof starts, must show without proof that they are tautologies.
6.1262 Proof in logic is only a mechanical expedient to facilitate
the recognition of tautology, where it is complicated.
----------------------------
(L. Wittgenstein, Tractatus Logico-Philosophicus, 1921)
F.
translogi
Wittgenstein again, and i even prommissed to write an essay on the
philosophical aspects of his logic....
(in only 3000 words I think i need that amount just to clarify were i
am talking about...)
Why did i ever enrol on Wittgenstein?
But thanks for the earlier reply how to prove it, i worked it out and
it has 10 assumptions and 39 lines.
Buugh
But Allan and Hand is a nice logic book but the proof of S49- S54 was
missing I now know why..
But again ~E is not a proper rule, i like intuitionistic logic.
.
But Frege <--> was not an connective in your logic I presumee?
I have to look in to your philosophy of logic as well.
(can you give me a short summary in 30.000 words or less?)
the prove as i worked it out:
~(P <-> Q) |- ~P <-> Q
1 (1) ~(P <-> Q) Premmisse
2 (2) ~P Assumption
3 (3) P Assumptiom
4 (4) ~Q Assumption
3,4 (5) P & ~Q 3 , 4 &I
3,4 (6) P 5 &E
2,3,4 (7) P & ~P 6 , 2 &I
2,3 (8) ~~Q 4 - 7 ~I
2,3 (9) Q 8 ~~E
2 (10) P --> Q 3 - 9 ->I
11 (11) ~Q Assumption
12 (12) Q Assumption
13 (13) ~P Assumption
12,13 (14) Q & ~P 12,13 &I
12,13 (15) Q 14 &E
11,12,13(16) Q & ~Q 15,11 &I
11,12 (17) ~~P 13-16 ~I
11,12 (18) P 17 ~~E
11 (19) Q --> P 12-18 ->I
2,11 (20) P <-> Q 10,19 <->I
1,2,11 (21) (P <-> Q) & ~(P <-> Q) 1, 20 &I
1,2 (22) ~~Q 11-21 ~I
1,2 (23) Q 22 ~~E
1 (24) ~P -> Q 2 -23 ->I
25 (25) Q Assumption
26 (26) P Assumption
25,26 (27) P & Q 25,26 &I
25,26 (28) Q 27 &E
25 (29) P --> Q 26-28 ->I
30 (30) P Assumption
31 (31) Q Assumption
30,31 (32) P & Q 30,31 &I
30,31 (33) P 32 &E
30 (34) Q --> P 31-33 ->I
25,30 (35) p<->Q 29,34 <->I
1,25,30 (36) (p <->Q) & ~(P <-> Q) 1 ,35 &I
1,25 (37) ~P 30-36 ~I
1 (38) Q -> ~P 25-37 ->I
1 (39) ~P <-> Q 24,38 <->I
I use an more explicit description in the (right) collumn, it is a
from Pospesel "propositopnal logic"
(an - after a line means that the assumption is discarded)
and I use ~~E instead of ~E (with i find inproper)
G. Frege
wrote:
>
> Wittgenstein again, and i even promised to write an essay on the
> philosophical aspects of his logic.
>
Hell, *I* wouldn't dare to do that. ;-)
>
> Why did i ever enroll on Wittgenstein?
>
Right. :-)
>
> But thanks for the earlier reply how to prove it, i worked it out and
> it has 10 assumptions and 39 lines.
>
Right. It seems that your system is very much like E. J. Lemmon's
system (in "Beginning Logic"). Though in Lemmon's system <-> is a
_defined_ symbol (hence no primitive rules for it).
Here's the proof done in Lemmon's system:
1 (1) ~(P <-> Q) A
2 (2) ~P A
3 (3) P A
4 (4) ~Q A
3,4 (5) P & ~Q 3,4 &I
3,4 (6) P 5 &E
2,3,4 (7) P & ~P 6,2 &I
2,3 (8) ~~Q 4,7 RAA
2,3 (9) Q 8 DN
2 (10) P -> Q 3,9 CP
11 (11) ~Q A
12 (12) Q A
13 (13) ~P A
12,13 (14) Q & ~P 12,13 &I
12,13 (15) Q 14 &E
11,12,13(16) Q & ~Q 15,11 &I
11,12 (17) ~~P 13,16 RAA
11,12 (18) P 17 DN
11 (19) Q -> P 12,18 CP
2,11 (20) (P -> Q) & (Q -> P) 10,19 &I
2,11 (21) P <-> Q 20 Def<->
1,2,11 (22) (P <-> Q) & ~(P <-> Q) 1,21 &I
1,2 (23) ~~Q 11,22 RAA
1,2 (24) Q 23 DN
1 (25) ~P -> Q 2,24 CP
26 (26) Q A
27 (27) P A
26,27 (28) P & Q 26,27 &I
26,27 (29) Q 28 &E
26 (30) P -> Q 27,29 CP
31 (31) P A
32 (32) Q A
31,32 (33) P & Q 31,32 &I
31,32 (34) P 33 &E
31 (35) Q -> P 32,34 CP
26,31 (36) (P -> Q) & (Q -> P) 30,35 &I
26,31 (37) P <-> Q 36 Def<->
1,26,31 (38) (P <-> Q) & ~(P <-> Q) 1,37 &I
1,26 (39) ~P 31,38 RAA
1 (40) Q -> ~P 26,39 CP
1 (41) (~P -> Q) & (Q -> ~P) 25,40 &I
1 (42) ~P <-> Q 41 Def<->
Rules:
A <---> Assumption
RAA <---> ~I
DN <---> ~~E (but also for ~~I)
CP <---> ->I
I personally prefer the name schema using "I" and "E". (Lemmon
obviously had some ancient rules from scholastic logic in mind when
giving names to his rules.)
>
> and I use ~~E instead of ~E (with i find improper)
>
Right. "~~E" is appropriate here.
>
> But <-> was not an connective in Frege's logic [in Begriffsschrift]
> I presume?
>
Right. In Begriffsschrift (1879) the only connectives were ~ and ->.
>
> I have to look in his philosophy of logic as well.
>
Good idea, and _much more_ rewarding than a look into Wittgenstein's
"philosophy of logic". W. is rather cryptic most of the times. :-)
G. Frege
wrote:
>
> C'mon, Charlie. Sock it to him.
>
No reply so far. :-(
F.
G. Frege
>>
>> and I use "~~E" instead of "~E" (with I find improper)
>>
> Right. "~~E" is appropriate here.
>
In Gentzen's original system of natural deduction, the names "~E" and
"~I" actually were appropriate. He had the rules:
A ~A
------- (~E)
_|_
and
[A]
:
_|_
----- (~I)
~A
Clearly "~I" and "~E" are appropriate names here.
Now in a system without "_|_" things are (have to be) slightly
different. Of course the following makes sense:
[A]
:
B & ~B
------ (~I)
~A
But what's with ~E?
My own idea concerning this question: In a system for classical
propositional logic we might adopt the rule
[~A]
:
B & ~B
------ (~E)
A
- though I'm not particularly lucky with it... :-/
From ~E we get immediately ~~E as a derived rule:
(1) ~~A Assumption
(2) ~A Assumption
(3) ~A & ~~A 1,2 &I
(4) A 2,3 ~E
In effect:
~~A
--- "~~E"
A
R. Srinivasan
> On 7 Feb 2007 05:13:03 -0800, "R. Srinivasan" <sradh...@in.ibm.com>
> wrote:
>
>
>
> > C'mon, Charlie. Sock it to him.
>
> No reply so far. :-(
>
generates these proofs. Don't know if it would work for your example,
though.
Regards, RS
George Dance
>
> How do you prove:
> ~(P <--> Q) |- ~P <--> Q
> using natural deduction?
Proof in a Copi-style system:
1. ~(P <-> Q)
2. ~((P & Q) v (~P & ~Q)) 1 Equivalence
3. (~(P & Q) & ~(~P & ~Q)) 2 DeMorgan's Rule
4. ~(P & Q) 3 Simplification
5. ~P v ~Q 4 DeM
6. ~Q v ~P 5 Commutation
7. ~~Q -> ~P 6 Double Negation
8. Q -> ~P 7 Implication
9. ~(~P & ~Q) 3 Simp
10. ~~P v ~~Q 9 DeM
11. ~~~P -> ~~Q 10 Impl
12. ~P -> Q 11 DN (on both sides)
13. (Q -> ~P) & (~P -> Q) 8, 12 Conjunction
14. (~P -> Q) & (Q -> ~P) 13 Comm
15. ~P <-> Q 14 Equiv
Depending on the given rules of the system you're learning, you may
have to split line 12 into two steps; or you may be able to skip line
13. But those are technical details.
>
> I knew there were some logic provers on the web but lost the links to
> them.
> can anybody suply them.
>
Maybe someone can, but I won't. Learn to be your own logic prover.
translogi
[~B}
|
V
A & ~A
________ ~E (RAA Reductio ad absurdum)
B
It is just that i like intuitionistic logic and this is NOT VALID in
intuitioniostic logic
[~B}
|
V
A & ~A
________ ~I
~~B
Is valid.
and
~~B
____ ~~E (DNE Double negatuion elimination)
B
is not valid in intuitionistic logic.
But at least then you know that it is invalid in intuitionistic logic
but valid in classical logic.
I am always puzzeling shall i use -->I or CP ~I or RAA
It is a bit if i am a mathematician ( likes -->I and ~I )
or the philosopher (those non descript symbold will not convince your
mother in Law so think of something more descriptive)
So it depends on what the purpose is.
Mathematics or Philosophics.
I still have to read Gentzens original.
it is claimed that he used an axiom for ~~E DNE.not a rule.
(no don't ask me what the fine difference between the two is)
I supposse _|_ is just shorthand for any contradiction (in classical
or intuitionistic logic)
or
Something like ~~P & ~P in (Johansens) minimal logic (P & ~P is not
the same in this logic)
But the term minimal logic is a bit misused, there are atleast two of
them
Back to Frege, Wittgenstein & Hantikke
G. Frege
wrote:
>
> ~E is used as
>
> [~B]
> |
> V
> A & ~A
> ------- ~E (RAA Reductio ad absurdum)
> B
>
*lol* Did you READ what I have written? :-)
>>
>> My own idea concerning this question: In a system for classical
>> propositional logic we might adopt the rule
>>
>> [~A]
>> :
>> B & ~B
>> ------ (~E)
>> A
>>
So your system (actually Pospesel's) just incorporated that idea! :-)
But, I also said that...
>>
>> ...I'm not particularly lucky with [this rule] ... :-/
>>
For the following reason:
>
> [...] this is NOT VALID in intuitionistic logic.
>
Right!
>
> [~B]
> :
> A & ~A
> ------ ~I
> ~~B
>
> is valid.
>
> and
>
> ~~B
> --- ~~E (DNE Double negation elimination)
> B
>
> is not valid in intuitionistic logic.
>
Right.
>
> But at least then you know that it is invalid in intuitionistic logic
> but valid in classical logic.
>
Right.
That's certainly an argument for using
~I and ~~E
instead of
~I and ~E
(in systems where there is no "_|_" available).
>
> I am always puzzling shall i use ->I or CP, ~I or RAA?
>
*lol* :-)
Right. Funny thing: Gentzen had ->I and ~I (as well as &I, vI, &E, &E,
etc. etc.) But _Lemmon_ decided to use CP and RAA instead of ->I and
~I. (I already mentioned that: it seems that he was influenced by
scholastic logic. For example "->E" for him is "MPP" (!), i.e. "modus
ponendo ponens". :-)
>
> It is a bit if i am a mathematician (likes ->I and ~I )
> or the philosopher (those non descript symbol will not
> convince your mother in Law so think of something more
> descriptive)
>
Right. --> "Lemmon's dilemma". :-)
>
> So it depends on what the purpose is.
> Mathematics or Philosophic.
>
Right. Good idea.
>
> I still have to read Gentzens original.
> it is claimed that he used an axiom for ~~E/DNE. not a rule.
>
Yes. When you add A v ~A as an axiom to the rules for the
intuitionistic calculus you get the classical calculus. Though he
mentions that an other possibility would be to add the rule
~~A
---
A
instead.
>
> (no don't ask me what the fine difference between the two is)
>
No essential difference.
Gentzen: "I considered this formulation to be the most 'natural' one."
Well, not really a _mathematical_ or _logical_ argument. ;-)
>
> I suppose _|_ is just shorthand for any contradiction (in classical
> or intuitionistic logic)
>
Not really. It's "the false proposition" - a primitive symbol of the
system. Of course, if you are dealing with semantics, then v(_|_) = f.
From a "logical point of view" I'm rather fond of that logical means.
Again, "your mother in Law" would prefer P & ~P instead, I guess...
;-)
>>
>> In Gentzen's original system(s) of natural deduction, the names "~E"
>> and "~I" actually were appropriate. He had the rules:
>>
>> A ~A
>> ------- (~E)
>> _|_
>>
>> and
>> [A]
>> :
>> _|_
>> ----- (~I)
>> ~A
>>
>> Clearly "~I" and "~E" are appropriate names here.
>>
In addition he has the rule:
_|_
---
A ,
with other words: EFQ.
G. Frege
wrote:
>>>
>>> C'mon, Charlie. Sock it to him.
>>>
>> No reply so far. :-(
>>
> I think Charlie-Boo has, or at least claims to have, an algorithm that
> generates these proofs. Don't know if it would work for your example,
> though.
>
Still, no reply.
This guy just loves to make unsubstantiated (i.e. nonsensical) claims.
Phil
infinitely many natural numbers, all finite, then there is a point on
the nn line that is preceeded by infinitely many nn's, and which
therefore must be infinite, since every nn on the nn line equals the
number of nn's preceeding it (plus or minus 1). I don't want to take up
too much of your time, but I know you were getting hammered with BS
about your own analysis of the situation, and I wondered how it turned
out? I bailed because I decided that until I published something about
the subject, I would hear nothing but mindless defensiveness, rather
than an honest discussion. Nothing published yet, however (I've been
working on 3 physics papers; also not published!). Anyway, I've started
reading "Prime Obsession," by John Derbyshire, and it got me thinking
about the general subject again. Kind of weird to read the old threads
after not really even thinking about the subject for three years!
Phil
To email me off-list, cut off my head.
George Dance
> Hi George, this is Phil, from a few years back, with the "if there are
> infinitely many natural numbers, all finite, then there is a point on
> the nn line that is preceeded by infinitely many nn's, and which
> therefore must be infinite, since every nn on the nn line equals the
> number of nn's preceeding it (plus or minus 1). I don't want to take up
> too much of your time, but I know you were getting hammered with BS
> about your own analysis of the situation, and I wondered how it turned
> out? I bailed because I decided that until I published something about
> the subject, I would hear nothing but mindless defensiveness, rather
> than an honest discussion.
To tell you the truth, I intervened only because there was no honest
discussion, and also bailed shortly after you did. It's sad how most
such 'discussions' turn into flamefests.
> Nothing published yet, however (I've been
> working on 3 physics papers; also not published!). Anyway, I've started
> reading "Prime Obsession," by John Derbyshire, and it got me thinking
> about the general subject again. Kind of weird to read the old threads
> after not really even thinking about the subject for three years!
>
Keep the problem in mind, but (to throw my 2 cents in for free) give
priority to your articles for publication. Once one cracks the peer-
reviewed journals, one gains a great deal of credibility in some
eyes.
As for the problem, I still see it as paradoxical. However, I think
that the root of the paradox is simply the idea that we want a natural
number for every possible quantity of actual existents. Assuming that
there are an infinite number of possible quantities, and that every
possible quantity of actual existents is finite, that leads
straightaway to the paradox: there must be an infinite number of
naturals, every one of which is finite.
The way I escape the paradox, in my own math, is to simply declare
that all the math I use employes only a finite subset of N, up to and
including google, which is paradox-free. If I ever need to use a
larger subset, I'll define that in term.
Phil
> On Feb 27, 1:18 am, Phil <toob-head...@sbcglobal.net> wrote:
>
>>Hi George, this is Phil, from a few years back, with the "if there are
>>infinitely many natural numbers, all finite, then there is a point on
>>the nn line that is preceeded by infinitely many nn's, and which
>>therefore must be infinite, since every nn on the nn line equals the
>>number of nn's preceeding it (plus or minus 1). I don't want to take up
>>too much of your time, but I know you were getting hammered with BS
>>about your own analysis of the situation, and I wondered how it turned
>>out? I bailed because I decided that until I published something about
>>the subject, I would hear nothing but mindless defensiveness, rather
>>than an honest discussion.
>
>
>
> To tell you the truth, I intervened only because there was no honest
> discussion, and also bailed shortly after you did. It's sad how most
> such 'discussions' turn into flamefests.
Well I must say, it was VERY nice, given that you had previously said
(if I remember correctly) that you thought I was in general confused, to
have you stand up and say that you couldn't figure out what was wrong
with this particular paradox, even though you thought that it probably
was wrong, somewhere. Thanks, I did appreciate that.
Unfortunately, as you say, people are not yet willing to have completely
open-minded, honest discussions on the internet, perhaps because in such
discussions, our beliefs become completely vulnerable, open to attack or
even destruction. Usually, we are simply not willing to risk losing our
beliefs, even if science states that we should, from time to time,
re-examine our beliefs in just such a vulnerable, open manner. Of
course, this applies just as much to the cranks as to the true-believers.
>
>
>
>>Nothing published yet, however (I've been
>>working on 3 physics papers; also not published!). Anyway, I've started
>>reading "Prime Obsession," by John Derbyshire, and it got me thinking
>>about the general subject again. Kind of weird to read the old threads
>>after not really even thinking about the subject for three years!
>>
>
>
>
> Keep the problem in mind, but (to throw my 2 cents in for free) give
> priority to your articles for publication. Once one cracks the peer-
> reviewed journals, one gains a great deal of credibility in some
> eyes.
Probably good advice! Any suggestions as to which journal(s) would be
good for this particular type of problem?
>
>
> As for the problem, I still see it as paradoxical. However, I think
> that the root of the paradox is simply the idea that we want a natural
> number for every possible quantity of actual existents. Assuming that
> there are an infinite number of possible quantities, and that every
> possible quantity of actual existents is finite, that leads
> straightaway to the paradox: there must be an infinite number of
> naturals, every one of which is finite.
>
Well ... I'm not completely certain I understand what you mean here. The
"infinite number of quantities" is only a problem if they are unique, if
no two of these quantities are the same, which is probably what you
mean, but yes, if we then assume or conclude that there are infinitely
many quantities, and further assume or conclude that every quantity is
finite, we have the current beliefs about natural numbers, and then the
paradox.
I found a solution that satisfied me a few weeks after I bailed, after
which I quit thinking about it. I concluded that we must be consciously
aware of whether our arguments use potential or actual infinity.
Mathematicians implicitly used actual infinity in their proof showing
that there are infinitely many numbers, but then implicitly used
potential infinity in the proof showing that all numbers are finite. Of
course, this is a problem ONLY if potential and actual infinity cannot
be "mixed and matched," if using both concepts to determine various
characteristics of some subject, such as the natural numbers, leads to
contradictions. My paradox is simply a demonstration that the answer is
that potential and actual infinity do lead to contradictions, that they
cannot both be used for a single situation, anymore than we can assume
that parallel lines both do, and do not, meet, in the same proof.
The four possibilities, given the two premises of potential (P) and
actual (A) infinity, are (1) A&A: infinitely many numbers, infinitely
many of which (but not all!) are infinite, (2) A&P: infinitely many
numbers, all of which are finite, (3) P&A: finitely many numbers, a
finite number of which are infinite, (4) P&P: finitely many numbers, all
of which are finite. Only #1 and #4 do not "mix and match" potential and
actual infinity, and can therefore be self-consistent.
> The way I escape the paradox, in my own math, is to simply declare
> that all the math I use employes only a finite subset of N, up to and
> including google, which is paradox-free. If I ever need to use a
> larger subset, I'll define that in term.
>
If my understanding is correct, your method should work fine, since it
is equivalent to #4 above. The only potential for error would be to use
something else that uses actual infinity, and I think that even then,
sometimes, this would not cause contradictions/paradoxes, depending on
what the "thing" is that uses actual infinity, and how it is used.
>
>
Oh well, I do NOT mean to drag you into another flamefest, so I'll let
it go at that. Also, I am absolutely not asking you to research a good
journal to publish this! I'm just hoping that you can think of one off
the top of your head.
G. Frege
>>>
>>> If there are infinitely many natural numbers, all finite, then there
>>> is a point on the nn line that is preceeded by infinitely many nn's,
>>> and which therefore must be infinite,
>>>
No.
>>>
>>> since every nn on the nn line equals the number of nn's preceeding it.
>>>
The "since" doesn't make sense here. It's just:
"Every nn on the nn line equals the number of nn's preceeding it."
Right. And since the number of nn's preceding any (every) point on the nn
line is finite, any (every) nn on the nn line is finite too.
Of course we might IMAGINE a "number line" with infinite ordinals on it.
Consider:
0 1 2 3 ... w w+1
|----|----|----|-- ... ---|---|---
Here the number of natural numbers preceeding w is NOT finite any more; and
(consequently) w is not a finite (i.e. natural) number! :-)
Note though that this number line is not the line of (all and only) natural
numbers (i.e. the nn line).
>>>
>>> I know you were getting hammered with BS about your own analysis of the
>>> situation ...
>>>
Right. This was justified since he just came up with nonsensical claims. I
wont say that he's an idiot (in general), but concerning mathematical
questions he is.
>>
>> As for the problem, I still see it as paradoxical.
>>
Where is the _paradox_? Imho you are mixing up (i.e. mis-take) a counter-
intuitive mathematical fact (which, I admit, it might be) with a logical
_paradox_. (Oh, sorry, it's not you but George Dance. Right. See my comment
above. :-)
>>
>> However, I think that the root of the paradox is simply the idea that
>> we want a natural number for every possible quantity of actual existents.
>>
Nonsense. Whatever GD is talking about here; it's not mathematics.
More reasonable: We want (at least in set theory) a natural number for
every finite set (as a measure of its "size", i.e. cardinality). :-)
>>
>> ...there must be an infinite number of naturals, every one of which
>> is finite.
>>
Right. So where's the paradox? :-o
>
> I found a solution ...
>
Well, how can you find a _solution_ if there is no _problem_ in the first
place? :-o
>
> that satisfied me a few weeks after I bailed, after
> which I quit thinking about it. I concluded that we must be consciously
> aware of whether our arguments use potential or actual infinity.
>
Well, this certainly might be an interesting question from a logical
(and/or philosophical) point of view.
>
> Mathematicians implicitly used actual infinity in their proof showing
> that there are infinitely many numbers, but then implicitly used
> potential infinity in the proof showing that all numbers are finite. Of
> course, this is [or would be] a problem ONLY if potential and actual
> infinity cannot be "mixed and matched," ...
>
Right. It can --- if we give the preference to /actual infinity/, as Cantor
himself already noted.
>>
>> The way I escape the paradox, in my own math ...
>>
Hahahaha!!! Good joke! :-)
George Dance
> On Mon, 05 Mar 2007 07:35:24 GMT, Phil <toob-head...@sbcglobal.net> wrote:
>
>
>
> >>> I know you were getting hammered with BS about your own analysis of the
> >>> situation ...
>
> Right. This was justified since he just came up with nonsensical claims. I
> wont say that he's an idiot (in general)
Thank you for illustrating Phil's point. Good soldier Fritsche can
always be counted on when BS is needed.
> but concerning mathematical
> questions he is.
>
> >> As for the problem, I still see it as paradoxical.
>
> Where is the _paradox_?
In the part you snipped. I suggest you go back and reread.
> Imho you are mixing up (i.e. mis-take) a counter-
> intuitive mathematical fact (which, I admit, it might be) with a logical
> _paradox_. (Oh, sorry, it's not you but George Dance. Right. See my comment
> above. :-)
>
Confusing logic with mathematics? Sorry, that wasn't me, but Bertrand
Russell. Perhaps you can spend the rest of the day flaming him
instead.
> >> However, I think that the root of the paradox is simply the idea that
> >> we want a natural number for every possible quantity of actual existents.
>
> Nonsense. Whatever GD is talking about here; it's not mathematics.
> More reasonable: We want (at least in set theory) a natural number for
> every finite set
Even Kant believed he could count his fingers. Sad how things have
deteriorated since his day.
> > that satisfied me a few weeks after I bailed, after
> > which I quit thinking about it. I concluded that we must be consciously
> > aware of whether our arguments use potential or actual infinity.
>
> Well, this certainly might be an interesting question from a logical
> (and/or philosophical) point of view.
>
But for someone like you, who has no interest in either a logical or a
philosophical point of view, it's boring. Which doesn't stop you
writing, alas.
>
> >> The way I escape the paradox, in my own math is to simply declare
that all the math I use employes only a finite subset of N, up to and
including google, which is paradox-free. If I ever need to use a
larger subset, I'll define that in term.
> Hahahaha!!! Good joke! :-)
>
Chris Menzel
said:
> ...
> Unfortunately, as you say, people are not yet willing to have
> completely open-minded, honest discussions on the internet, perhaps
> because in such discussions, our beliefs become completely vulnerable,
> open to attack or even destruction. Usually, we are simply not willing
> to risk losing our beliefs, even if science states that we should,
> from time to time, re-examine our beliefs in just such a vulnerable,
> open manner. Of course, this applies just as much to the cranks as to
> the true-believers.
Unfortunately, often as not, cranks who can't follow a simple
mathematical argument, or who are just hopelessly confused, misinterpret
their interlocutors' unwillingless to acknowledge the "soundness" of
their cranky arguments as proof that their interlocutors are simply
dogmatic true believers who don't want to risk losing their beliefs.
G. Frege
wrote:
>>
>> [...] This was justified since he just came up with nonsensical claims.
>>
> Thank you for illustrating Phil's point.
>
But what can I do, George? It's just the truth! :-)
>>
>> Imho you are mixing up (i.e. mis-take) a counter-intuitive mathematical
>> fact (which, I admit, it might be) with a logical _paradox_.
>>
> Confusing logic with mathematics? Sorry, that wasn't me, but Bertrand
> Russell. Perhaps you can spend the rest of the day flaming him ...
>
No need to do that. One just might read Gödel's article "Russell's
Mathematical Logic" instead.
>>
>> More reasonable: We want (at least in set theory) a natural number for
>> every finite set
>>
> Even Kant believed he could count his fingers. Sad how things have
> deteriorated since his day.
>
Well, actually in axiomatic set theory (say ZFC) _everything is set_ (hence
no fingers there).
>>>
>>> I concluded that we must be consciously aware of whether our arguments
>>> use potential or actual infinity.
>>>
>> Well, this certainly might be an interesting question from a logical
>> (and/or philosophical) point of view.
>>
> But for someone who has no interest in either a logical or a
> philosophical point of view, it's boring.
>
Well, *I* don't know. :-)
Phil
belief, in order to determine whether any errors exist, we simply state,
oever and over, the belief in question. That "refutes" any possible
errors brought to light in the re-examination. You know, I think I'll
try that on my bank account. It DOES have $10,000, it REALLY DOES have
$10,000, any IDIOT KNOWS that it has $10,000... According to you, it
should now have $10 K! Thanks! No need to examine the criticisms, or the
paradox, or the bank statement. And here I thought that the filibuster
method was mainly a crank method.
I don't suppose you could come up with a logical train of thought that
shows why we CAN mix-and-match potential and actual infinity? That would
save me the time of writing it up, and I will publicly thank you and
acknowledge your superior intellect if you do so. Remember, so I don't
waste YOUR time, I have no disagreement with either the argument proving
infinitely many numbers, nor the argument proving they are all finite.
It's the combination that I think violates the rules of mathematics,
specifically the one that says we cannot combine results from two
mutually exclusive premises (in this case, potential and actual infinity).
I just want to make sure that (1) using potential infinity in the
argument proving infinitely many numbers will indeed now prove finitely
many numbers (or if you prefer, "potentially infinite numbers"), (2)
using actual infinity in the argument proving every number is finite
will now prove that some (infinitely many but not all) are infinite
"numbers" (or however we refer to such beasts), and (3) that using
"mix-and-match" systems, such as, there are finitely many numbers, some
of which are infinite" -- or its inverse, the current beliefs for
natural numbers -- produces contradictions/paradoxes.
Thanks in advance for giving a valid correction, and not merely a
"filibuster" correction!
But who am I kidding? As George says, the only option here is to
publish, so that I won't have to listen to idiotic "responses." What's
really going to piss me off then is that these same idiots, who are now
blasting me, will then blast anyone who dares to question the latest
"Holy Belief" of the "Holy Experts," namely that there CANNOT be
infinitely many numbers, all of which are finite. Then, the comments
will be, "Who could be stupid enough to think that a finite number of
digits -- necessary, if all numbers are finite -- could possibly be used
to create infinitely many numbers???" Sheesh ...
Phil
Chris Menzel
said:
> Chris Menzel wrote:
>> On Mon, 05 Mar 2007 07:35:24 GMT, Phil <toob-h...@sbcglobal.net>
>> said:
>>>...
>>>Unfortunately, as you say, people are not yet willing to have
>>>completely open-minded, honest discussions on the internet, perhaps
>>>because in such discussions, our beliefs become completely
>>>vulnerable, open to attack or even destruction. Usually, we are
>>>simply not willing to risk losing our beliefs, even if science states
>>>that we should, from time to time, re-examine our beliefs in just
>>>such a vulnerable, open manner. Of course, this applies just as much
>>>to the cranks as to the true-believers.
>>
>> Unfortunately, often as not, cranks who can't follow a simple
>> mathematical argument, or who are just hopelessly confused, misinterpret
>> their interlocutors' unwillingless to acknowledge the "soundness" of
>> their cranky arguments as proof that their interlocutors are simply
>> dogmatic true believers who don't want to risk losing their beliefs.
>>
> Ah yes, the "filibuster logical proof" method.
What are you talking about? I didn't even mention your argument (I'm
not even sure what it was) or accuse you personally of being a crank.
Nor did I assert anything that was in need of any proof beyond lurking
in sci.logic/math/physics for a couple of weeks. I was simply balancing
your quite valid point about the importance of being open to
re-examining our own beliefs with a simple observation about genuine
cranks, viz., their unwillingness or inability to see their own
elementary confusions and, as a consequence, to accuse those who try to
point them out (often helpfully and, granted, sometimes abusively) of
being irrationally blinded by their own dogmas or driven by their thirst
for power and recognition, whatever. Your, um, agitated response to
this simple empirical observation was really quite bizarre.
george
> Chris Menzel wrote:
> > Unfortunately, often as not, cranks who can't follow a simple
> > mathematical argument, or who are just hopelessly confused, misinterpret
> > their interlocutors' unwillingless to acknowledge the "soundness" of
> > their cranky arguments as proof that their interlocutors are simply
> > dogmatic true believers who don't want to risk losing their beliefs.
On Mar 5, 1:01 pm, Phil <toob-head...@sbcglobal.net> wrote:
> Ah yes, the "filibuster logical proof" method. When re-examining some
> belief, in order to determine whether any errors exist, we simply state,
> oever and over, the belief in question. That "refutes" any possible
> errors brought to light in the re-examination.
Oh, shut up.
Purported proofs that are not actually proofs
ARE ALWAYS EASY to refute. ALWAYS.
BY DEFINITION.
Proof-predicates are PRIMITIVE recursive.
The ONLY thing that make a proof HARD is to make it LONG.
If the proof is SHORT then checking it IS *ALWAYS* easy.
This is just a fact about the way proofs are DEFINED.
It is, I repeat, A FACT (because it is a definition).
Therefore, YOU don't GET to have an OPINION about this matter.
If you think you have a proof, then state it. If it is not a proof,
then the reason why WILL NOT depend on anybody filibustering
anything. It will just be that it does not match any correct
SYNTACTIC
FORM of a proof. That is the ONLY mode of refutation that is
possible
or relevant. Neither filibustering NOR "repeated assertions" of
ANYthing
will have ANYthing to do with it. So forget the sociology of
persecution
and CONFINE yourself to presenting your math.
> What's
> really going to piss me off then is that these same idiots, who are now
> blasting me, will then blast anyone who dares to question the latest
> "Holy Belief" of the "Holy Experts," namely that there CANNOT be
> infinitely many numbers, all of which are finite.
You've got your who's who's BACKWARDS, here.
The people you are maligning as witch doctors are the ones who
KNOW that there CAN be (and in fact are) infinitely many numbers,
all of which are finite. Nobody knows anybody who is disputing that
coherently, and most of us don't know anybody who is disputing it
at all. Are YOU disputing it? WHICH side are YOU on?
G. Frege
>
> I don't suppose you could come up with a logical train of thought that
> shows why we CAN mix-and-match potential and actual infinity?
>
Well, actually Cantor wrote something about that question!
His argument in a nutshell: The notion of potential infinity only makes
sense on the "basis" of [the notion of] actual infinity. (But even if he's
wrong concerning this question, clearly the acceptance of "actual infinity"
allows for arguments referring to "potential infinity". The problem is just
the other way round. :-)
>
> Remember, so I don't waste YOUR time, I have no disagreement with either
> the argument proving infinitely many numbers, nor the argument proving
> they are all finite. It's the combination that I think violates the rules
> of mathematics,
>
It doesn't. There are infinitely many natural numbers, but each and any is
finite.
I'll agree that this MIGHT _sound_ strange. Maybe the following formulation
is easier to digest:
Each and any natural number is finite.
But the set of _all_ natural numbers is not bound.
(I.e. it does not have an upper boundary.) With other
words, there is no biggest natural number. (I guess
that's immediately clear, no? If W would be this number,
W+1 would be another natural number, but bigger. Which
is an aburdity.)
Phil
Oops! My mistake. I thought you meant that I was just a crank, who could
not follow the simple arguments that prove (1) there are infinitely many
natural numbers, and the equally simple argument (2) that they are all
finite, and that simply restating these arguments was your method of
"proving" that my paradox was flawed. Gee, think I'm a bit excessively
defensive??? I apologize, that was very ill-mannered of me.
Anyway, briefly, the argument was that if there are infinitely many
numbers, then there must exist at least one number x (actually,
infinitely many numbers), such that there are infinitely many numbers
preceeding x on the natural number line. Conversely, if there is no such
number x, then there cannot be infinitely many numbers on the nn line,
which is a contradiction, since the nn line by definition contains all
the natural numbers (unless, of course, like George F., you think that
the nn line is "mystically" limited by potential infinity, even though
the natural numbers themselves have access to actual infinity, making it
possible for infinitely many of them to exist). However, for any number
y, the number of numbers preceeding y is (y -1), and if we switch to the
whole number line, the number of numbers preceeding y is simply y. In
other words, y = the number of numbers preceeding it (minus 1 on the nn
line), so if there are infinitely many numbers preceeding y, then y MUST
be infinite, which of course contradicts the current beliefs. In
contrast, if there are infinitely many numbers, infinitely many of which
are infinite, this paradox dissappears, as is the case if there are
finitely many numbers, all of which are finite. In both of these
alterntives, we use either use actual infinity, or potential infinity,
for BOTH the number of numbers, and their maximum value.
I should mention that even George Dance thinks the paradox is flawed
somehow, that the arguments showing that there are infinitely many
numbers, all of which are finite, are not only valid in themselves
(which I agree with), but that they do not rest on incompatible
premises, the only different being that he's pretty much the only one
who both believes that, but who is honest enough to say that he doesn't
know why the paradox is wrong.
Bizarre Phil
David Marcus
> Anyway, briefly, the argument was that if there are infinitely many
> numbers, then there must exist at least one number x (actually,
> infinitely many numbers), such that there are infinitely many numbers
> preceeding x on the natural number line.
Why?
> Conversely, if there is no such
> number x, then there cannot be infinitely many numbers on the nn line,
> which is a contradiction, since the nn line by definition contains all
> the natural numbers (unless, of course, like George F., you think that
> the nn line is "mystically" limited by potential infinity, even though
> the natural numbers themselves have access to actual infinity, making it
> possible for infinitely many of them to exist). However, for any number
> y, the number of numbers preceeding y is (y -1), and if we switch to the
> whole number line, the number of numbers preceeding y is simply y. In
> other words, y = the number of numbers preceeding it (minus 1 on the nn
> line), so if there are infinitely many numbers preceeding y, then y MUST
> be infinite, which of course contradicts the current beliefs. In
> contrast, if there are infinitely many numbers, infinitely many of which
> are infinite, this paradox dissappears, as is the case if there are
> finitely many numbers, all of which are finite. In both of these
> alterntives, we use either use actual infinity, or potential infinity,
> for BOTH the number of numbers, and their maximum value.
--
David Marcus
Phil
>>Chris Menzel wrote:
>>
>>>Unfortunately, often as not, cranks who can't follow a simple
>>>mathematical argument, or who are just hopelessly confused, misinterpret
>>>their interlocutors' unwillingless to acknowledge the "soundness" of
>>>their cranky arguments as proof that their interlocutors are simply
>>>dogmatic true believers who don't want to risk losing their beliefs.
>
>
> On Mar 5, 1:01 pm, Phil <toob-head...@sbcglobal.net> wrote:
>
>>Ah yes, the "filibuster logical proof" method. When re-examining some
>>belief, in order to determine whether any errors exist, we simply state,
>>oever and over, the belief in question. That "refutes" any possible
>>errors brought to light in the re-examination.
>
>
> Oh, shut up.
> Purported proofs that are not actually proofs
> ARE ALWAYS EASY to refute. ALWAYS.
> BY DEFINITION.
Now wait a second. Proofs that contain deeply hidden errors are NOT
always easy to disprove. That is so obvious that I think we're talking
about two different things. I'm saying that when offering an analysis of
some current belief, that simply repeating the current beliefs is not a
valid refutation of that analysis, and we often saw just that kind of
"refutation" in the past on this subject.
> Proof-predicates are PRIMITIVE recursive.
> The ONLY thing that make a proof HARD is to make it LONG.
> If the proof is SHORT then checking it IS *ALWAYS* easy.
If people at some time use assumptions/premises that contain errors,
then making a proof short cannot possibly protect anyone from those
errors. Again, we HAVE to be talking about two different things, because
I know ou know better than to think that all easy to follow proofs MUST
be free of any errors. After all, isn't the Achilles an easy to follow
proof that we can never leave the room?
> This is just a fact about the way proofs are DEFINED.
> It is, I repeat, A FACT (because it is a definition).
> Therefore, YOU don't GET to have an OPINION about this matter.
> If you think you have a proof, then state it. If it is not a proof,
> then the reason why WILL NOT depend on anybody filibustering
> anything. It will just be that it does not match any correct
> SYNTACTIC
> FORM of a proof. That is the ONLY mode of refutation that is
> possible
> or relevant. Neither filibustering NOR "repeated assertions" of
> ANYthing
> will have ANYthing to do with it. So forget the sociology of
> persecution
> and CONFINE yourself to presenting your math.
Whatever.
>
>
>>What's
>>really going to piss me off then is that these same idiots, who are now
>>blasting me, will then blast anyone who dares to question the latest
>>"Holy Belief" of the "Holy Experts," namely that there CANNOT be
>>infinitely many numbers, all of which are finite.
>
>
> You've got your who's who's BACKWARDS, here.
> The people you are maligning as witch doctors are the ones who
> KNOW that there CAN be (and in fact are) infinitely many numbers,
> all of which are finite. Nobody knows anybody who is disputing that
> coherently, and most of us don't know anybody who is disputing it
> at all. Are YOU disputing it? WHICH side are YOU on?
>
Yes George, I have always been disputing the "fact" that there are
infinitely many numbers, all of which are finite, from the moment I
heard it, mainly because there is NO WAY that a finite number of digits
can be assembled into infinitely many unique numbers. In other words,
10^x, where x is finite, is itself finite, since it is simply 10 * 10 *
10 * ... * 10, with a FINITE number of 10's. You cannot multiply 10 by
itself a FINITE number of times and get an INFINITE "number," period.
Duh. And if every number is finite, then every number has a FINITE
number of digits. Also duh.
The number line paradox (and the 1/2 + 1/4 + ... versus 1/z, where z is
an infinitesimal, paradox) is simply a demonstration that something is
wrong with the belief that there are infinitely many numbers, all of
which are finite. Since the argument proving that there are infinitely
many numbers is fine, no flaws, and the argument proving that all
numbers are finite is fine, also no flaws, it MUST be the case that
either (1), the number line paradox is a proof with a flaw that is hard
to find, or (2) the two arguments use different, and incompatible,
premises. I say it's the latter, that the argument proving infinitely
many numbers makes use of actual infinity -- if it used potential
infinity, it would generate potentially many numbers -- and that the
argument proving that all numbers are finite uses potential infinity,
because if it used actual infinity, it would generate numbers with
infinite magnitude. You can combine two arguments if BOTH of them use
potential infinity, or BOTH use actual infinity, and get valid results,
but you cannot combine an argument that uses actual infinity with an
argument that uses potential infinity and get valid results, as both the
number line paradox, and the fact that 10 multiplied by itself a finite
number of times cannot produce an infinite result, demonstrate. Okay,
that's not a duh, but it should be clear, and that's what I intend to
publish (although hopefully in a prettier form, one that's more
acceptable to modern mathematicians).
Phil
Phil
> Phil wrote:
>
>>Anyway, briefly, the argument was that if there are infinitely many
>>numbers, then there must exist at least one number x (actually,
>>infinitely many numbers), such that there are infinitely many numbers
>>preceeding x on the natural number line.
>
>
> Why?
Read a little further ...
>
>
>>Conversely, if there is no such
>>number x, then there cannot be infinitely many numbers on the nn line,
>>which is a contradiction, since the nn line by definition contains all
>>the natural numbers (unless, of course, like George F., you think that
>>the nn line is "mystically" limited by potential infinity, even though
>>the natural numbers themselves have access to actual infinity, making it
>>possible for infinitely many of them to exist).
There you go. To emphasize a bit, if EVERY number on the nn line is
preceeded by a finite number of numbers, then there MUST be a finite
number of numbers on the nn line. Or, if there is no such thing as a
number which is preceeded by infinitely many numbers, then there CANNOT
POSSIBLY be infinitely many numbers on that particular line. Now, if you
want to argue this one, we're done. Why? Besides the obvious, because we
can put the numbers preceeding y into a set, and if there is no y such
that we can make a set with infinitely many elements, then there cannot
be infinitely many elements on the line. It's stupid having to even
discuss this, but I'm used to it. Instead of honestly asking whther
there is something useful here, most people -- maybe not you -- will
simply go into "stupid arguments" mode. Pathetic.
Phil
Phil
George, this looks useful and workable; I'll get back to it later!
Phil
David Marcus
What's an example of a stupid argument?
Are you saying that if there are infinitely many natural numbers, then
there must be a natural number y such that there are infinitely many
natural numbers less than y?
--
David Marcus
G. Frege
>
> ... briefly, the argument was that if there are infinitely many
> numbers, then there must exist at least one number x (actually,
> infinitely many numbers), such that there are infinitely many numbers
> preceeding x on the natural number line.
>
This is not an _argument_, but a _claim_. And it is wrong. (Actually, it's
a variant of Mückenheim's Axiom 1! :-)
On the other hand, if we wouldn't consider the nn line, but a "line" also
containing infinite ordinals (which are NOT natural numbers by definition)
0 1 2 3 ... w w+1 w+2
|----|----|----|- ... -|----|----|-- .
then your claim would actually be correct.
>
> Conversely, if there is no such number x, then there cannot be infinite-
> ly many numbers on the nn line,
>
Again, wrong.
(If there wouldn't be infinitely many natural numbers on the line, there
should be a biggest one, say X. Now is there any logical or mathematical
reason why X+1 should NOT be on the line?)
If we only consider the (finite) ordinal numbers proceeding the (infinite)
ordinal w, we have
0 1 2 3 ...
|----|----|----|- ...
i.e. we have infinitely many finite numbers.
David Marcus
> On Mon, 05 Mar 2007 22:14:19 GMT, Phil <toob-h...@sbcglobal.net> wrote:
> >
> > ... briefly, the argument was that if there are infinitely many
> > numbers, then there must exist at least one number x (actually,
> > infinitely many numbers), such that there are infinitely many numbers
> > preceeding x on the natural number line.
> >
> This is not an _argument_, but a _claim_. And it is wrong. (Actually, it's
> a variant of Mückenheim's Axiom 1! :-)
We could probably "prove" it from WM's Axiom 1!
--
David Marcus
G. Frege
<David...@alumdotmit.edu> wrote:
>>>
>>> ... briefly, the argument was that if there are infinitely many
>>> numbers, then there must exist at least one number x (actually,
>>> infinitely many numbers), such that there are infinitely many numbers
>>> preceeding x on the natural number line.
>>>
>> This is not an _argument_, but a _claim_. And it is wrong. (Actually, it's
>> a variant of Mückenheim's Axiom 1! :-)
>
> We could probably "prove" it from WM's Axiom 1!
>
Well, this shouldn't be too hard:
"Mückenheim Axiom 1:
X is not finite -> there must be an x in X which is infinite."
(F. Neugebauer, sci.math)
Phil
Yes.
>
David Marcus
Do you have a reason for saying this?
--
David Marcus
Phil
David Marcus
I guess that's a "no".
--
David Marcus
Phil
> On Mon, 05 Mar 2007 22:14:19 GMT, Phil <toob-h...@sbcglobal.net> wrote:
>
>
>>... briefly, the argument was that if there are infinitely many
>>numbers, then there must exist at least one number x (actually,
>>infinitely many numbers), such that there are infinitely many numbers
>>preceeding x on the natural number line.
>>
>
> This is not an _argument_, but a _claim_. And it is wrong. (Actually, it's
> a variant of Mückenheim's Axiom 1! :-)
>
> On the other hand, if we wouldn't consider the nn line, but a "line" also
> containing infinite ordinals (which are NOT natural numbers by definition)
>
> 0 1 2 3 ... w w+1 w+2
> |----|----|----|- ... -|----|----|-- .
>
> then your claim would actually be correct.
Well it certainly is true that on this line, each number on the line has
a value that is equal to the number of "numbers" (not necessarily
natural numbers) preceeding it.
>
>
>>Conversely, if there is no such number x, then there cannot be infinite-
>>ly many numbers on the nn line,
>>
>
> Again, wrong.
Oh really? So even if not one single number on a particular number line
is preceeded by infinitely many numbers, there could be infinitely many
numbers on that number line? Nope.
>
> (If there wouldn't be infinitely many natural numbers on the line, there
> should be a biggest one, say X.
This isn't true, but it's irrelevant to my argument, so I'm not going to
spend any real time on it. The fact that a sequence of objects falls
under the category of potential infinity, rather than actual infinity,
does NOT make it possible to specify "a biggest one," at least not in
the manner you are doing. If we look at the Achilles from the viewpoint
of potential infinity, what is the biggest number of halfway points that
can exist? There is no specifiable number, and yet as long as we cross
those halfway points at a finite rate (of halfway pionts) per minute for
a finite period of time, we will not leave the room, nor will we cross
an infinite number of points. Also, if you DEFINE X as being "the
biggest one," then adding one to it REDEFINES X as being (X-1), an
element which is one smaller than "the largest element," which then
tends to cause a "shift-error," drawing a conclusion that technicaly
applies to (X-1) -- namely, that it isn't the largest element -- and
mistakenly applying it to X. In any case, that's a side issue.
Now is there any logical or mathematical
> reason why X+1 should NOT be on the line?)
>
> If we only consider the (finite) ordinal numbers proceeding the (infinite)
> ordinal w, we have
>
> 0 1 2 3 ...
> |----|----|----|- ...
>
> i.e. we have infinitely many finite numbers.
And we agree, there are infinitely many numbers on the natural number
line. However, you should not assume that which you want to prove. Given
a criticism of the belief that there are infinitely many numbers, all of
which are finite, you must not ASSUME that there are infinitely many
numbers, all of which are finite, and use that assumption to "prove"
that the criticism is "therefore" false. The question remains, given
that there MUST be numbers on the line preceeded by infinitely many
numbers -- in fact, there must be infinitely many such numbers, since
infinite minus finite equals infinite -- how can such numbers not have
values which are equal to the number of numbers (within 1) preceeding
them, which is the case if all the numbers are finite? That is a
contradiction, one which results from the belief that there are
infinitely many numbers, all of which are finite, and which cannot be
properly eliminated by simply ASSUMING that there are infinitely many
numbers, all of which are finite.
Normally, such contradictions indicate one of two things; (1) there is
an error in the contradiction, or (2) there is an error in the premises,
or combination of premises, leading to the contradiction. If mathematics
is to be honest, rather than dogmatic, then we must make an HONEST
effort to determine which of these two possibilities is the case, rather
than make a concerted effort to "prove" that the contradiction is false,
false, FALSE I SAY!!!
Picky Phil
>
>
> F.
>
Phil
Oh no, as soon as you start dealing me straight, stop pretending that I
haven't given you several reasons already, and stop acting like a litle
boy, with an unending series of "why" "why" "why" regardless of what I
say, I'll deal you straight. But not until then. That's what happens
when you're useless.
David Marcus
I haven't seen even one reason, much less several. All I've seen is you
keep repeating the above statement. So, what's your proof of the above
statement? Do you have one?
--
David Marcus
George Dance
> On Mar 5, 3:23 am, G. Frege <nomail@invalid> wrote:
> > >>> I know you were getting hammered with BS about your own analysis of the
> > >>> situation ...
>
> > > Right. This was justified since he just came up with nonsensical claims. I
> > > wont say that he's an idiot (in general)
>
> > Thank you for illustrating Phil's point. Good soldier Fritsche can
> > always be counted on when BS is needed. </us>
>
>
> But what can I do, George?
Taking your meds might help.
George Dance
> On Mon, 05 Mar 2007 18:01:20 GMT, Phil <toob-head...@sbcglobal.net> wrote:
>
> > I don't suppose you could come up with a logical train of thought that
> > shows why we CAN mix-and-match potential and actual infinity?
>
> Well, actually Cantor wrote something about that question!
>
> His argument in a nutshell: The notion of potential infinity only makes
> sense on the "basis" of [the notion of] actual infinity.
That's not an argument; it's a claim. Does Cantor give an argument?
George Dance
>
>
> > It doesn't. There are infinitely many natural numbers, but each and any is
> > finite.
>
> > I'll agree that this MIGHT _sound_ strange. Maybe the following formulation
> > is easier to digest:
>
> > Each and any natural number is finite.
>
> > But the set of _all_ natural numbers is not bound.
> > (I.e. it does not have an upper boundary.) With other
> > words, there is no biggest natural number. (I guess
> > that's immediately clear, no? If W would be this number,
> > W+1 would be another natural number, but bigger. Which
> > is an aburdity.)
>
> > F.
>
> George, this looks useful and workable; I'll get back to it later!
>
His name isn't George, BTW. He uses the name "G. Frege" (from Gottlob
Frege, the inventor of predicate logic) as an alias; for all anyone
knows, he may believe that he is Gottlob Frege.
.
Phil
50, I have an incurable type of cancer called follicular non-Hodgkin's
lymphoma that has an average life expectancy of 7 years, and I've had it
7 years. Older males -- I assume you are a young man -- have a rule of
thumb, which they never say to anyone. As soon as they figure out that
person X is going to play debate games, instead of having honest
discussions, or repeat things told in private, or similar things, they
cut off person X, without ever telling person X. I don't have time to
play games, literally. I live on hydrocodone as it is, have chemo-brain,
and I simply don't have time for it. Your insistence on pretending to
not notice what I have written, to play dumb, to play con games, makes
you not just useless to me, but downright creepy. Fuck off, creep.
Phil
Oh, thanks! I was just thinking we have too damn many "Georges" around
here to keep track of, and that will help!
Chris Menzel
said:
> ...
> Anyway, briefly, the argument was that if there are infinitely many
> numbers, then there must exist at least one number x (actually,
> infinitely many numbers), such that there are infinitely many numbers
> preceeding x on the natural number line.
I'm afraid I don't see an argument there, only an assertion, moreover
one that is easily show to be false in set theory. More exactly, it is
easy to prove in set theory that (1) there are infinitely many natural
numbers, and (2) for any natural number n, there are only finitely many
natural numbers preceding n. Is it your claim that there is a flaw in
the usual proofs of these theorems of set theory? If so, what is the
flaw?
Phil
> On Mon, 05 Mar 2007 18:01:20 GMT, Phil <toob-h...@sbcglobal.net> wrote:
>
>
>>I don't suppose you could come up with a logical train of thought that
>>shows why we CAN mix-and-match potential and actual infinity?
>>
>
> Well, actually Cantor wrote something about that question!
>
> His argument in a nutshell: The notion of potential infinity only makes
> sense on the "basis" of [the notion of] actual infinity. (But even if he's
> wrong concerning this question, clearly the acceptance of "actual infinity"
> allows for arguments referring to "potential infinity". The problem is just
> the other way round. :-)
I'm not saying the two are unrelated. I'm saying that the reason
mathematicians are currently claiming things like, 10 to a FINITE power
is infinite -- which must be true if there are indeed infinitely many
numbers, all of which are finite -- is because using potential infinity
or actual infinity as a premise has certain consequences, and trying to
combine an argument based on P-oo with an argument based on A-oo
sometimes produces invalid, contradictory results, which is WHY the
natural number line paradox exists.
>
>
>>Remember, so I don't waste YOUR time, I have no disagreement with either
>>the argument proving infinitely many numbers, nor the argument proving
>>they are all finite. It's the combination that I think violates the rules
>>of mathematics,
>>
>
> It doesn't. There are infinitely many natural numbers, but each and any is
> finite.
That is the current claim. If true, it should not lead to true
paradoxes, only apparent ones. We cannot prove that the number line
paradox is only an apparent paradox by CLAIMING that "there are
infinitely many numbers, all of which are finite." We have to either
just trust in the wisdom of the Great Ones -- which is actually not that
bad of a thing to do -- or find a legitimate error. Note, however, that
if the current beliefs are wrong, we will have helped mathematics, not
hurt it, by revealing a long standing error. And there are errors to be
found. It's just a question of how many errors can be found by ordinary
minds (like mine), and whether this is one of them.
>
> I'll agree that this MIGHT _sound_ strange. Maybe the following formulation
> is easier to digest:
>
> Each and any natural number is finite.
>
> But the set of _all_ natural numbers is not bound.
> (I.e. it does not have an upper boundary.) With other
> words, there is no biggest natural number. (I guess
> that's immediately clear, no? If W would be this number,
> W+1 would be another natural number, but bigger. Which
> is an aburdity.)
Being unbound gives you EITHER potential or actual infinity. In the
Achilles, the number of halfway points is unbound, but if we cross them
"using" potential infinity, we will not leave the room in a finite
period of time, whereas if we cross them using actual infinity, we have
the possibility (not guaranteed, as there are several actual infinities)
of leaving the room, and doing so in a finite period of time. Note that
simply declaring that the halfway points are "unbound" does not tell us
which outcome will occur. For that, we need to know whether potential or
actual infinity is involved. Similarly, stating that the natural numbers
are unbound does not tell us whether they are potentially infinite or
actually infinite in either quantity or magnitude. However, adding that
all natural numbers are finite (in magnitude) DOES tell us the answer,
because if we try to say there are infinitely many numbers, we start
creating paradoxes, i.e., contradictions (I know of three, but I am sure
there are many more).
Phil
>
>
> F.
>
David Marcus
Why assume that? But then, why assume there must be a natural number
greater than an infinite number of natural numbers? I guess if you are
going to assume things, the sky is the limit.
> -- have a rule of
> thumb, which they never say to anyone. As soon as they figure out that
> person X is going to play debate games, instead of having honest
> discussions, or repeat things told in private, or similar things, they
> cut off person X, without ever telling person X. I don't have time to
> play games, literally. I live on hydrocodone as it is, have chemo-brain,
> and I simply don't have time for it. Your insistence on pretending to
> not notice what I have written, to play dumb, to play con games, makes
> you not just useless to me, but downright creepy. Fuck off, creep.
I've read your recent posts. You said that there must be a natural
number which is greater than an infinite number of natural numbers.
However, you haven't offered any sort of proof or reason, although you
apparently think you have. Plus, your statement is clearly false, of
course.
Is your medical problem relevant to your mathematical statement?
--
David Marcus
G. Frege
wrote:
>>
>> But what can I do, George?
>>
> Taking your meds might help.
>
Can't stand the truth? :-)
David Marcus
> I'm not saying the two are unrelated. I'm saying that the reason
> mathematicians are currently claiming things like, 10 to a FINITE power
> is infinite
I can't say any mathematicians I know have ever claimed that. Do you
have a reference?
> -- which must be true if there are indeed infinitely many
> numbers, all of which are finite
How does this imply the preceding?
> However, adding that
> all natural numbers are finite (in magnitude) DOES tell us the answer,
> because if we try to say there are infinitely many numbers, we start
> creating paradoxes, i.e., contradictions (I know of three, but I am sure
> there are many more).
What are the three?
--
David Marcus
G. Frege
wrote:sensecompaprocessespotential
>>
>> On the other hand, if we wouldn't consider the nn line, but a "line" also
>> containing infinite ordinals (which are NOT natural numbers by definition)
>>
>> 0 1 2 3 ... w w+1 w+2
>> |----|----|----|- ... -|----|----|-- .
>>
>> then your claim would actually be correct.
>
> Well it certainly is true that on this line, each number on the line
> has a value that is equal to the number of "numbers" (not necessarily
> natural numbers) preceeding it.
>
Right. But it's also true for the following line (i.e. the nn line):
0 1 2 3 ...
|----|----|----|- ...
--- though THIS line does only contain natural numbers (by definition).
>>
>> [You are] wrong.
>>
> Oh really? So even if not one single number on a particular number line
> is preceeded by infinitely many numbers, there could be infinitely many
> numbers on that number line?
>
Exactly.
You might consider the following argument:
>>
>> If there wouldn't be infinitely many natural numbers on the line, there
>> should be a biggest one, say X.
>>
> This isn't true.
>
OF COURSE this is true.
Since otherwise (if there wouldn't be a biggest natural number on the line)
the natural numbers on the line would not have an upper bound, and hence
there would be infinitely many natural number on the line.
>
> but it's irrelevant to my argument ...
>
Of course it's NOT irrelevant to your "argument" --- since it PROVES
["prove" used in a nontechnical sense here] you wrong.
Let me repeat the final sentence of my argument:
>
> But there is is no logical or mathematical reason why X+1 should NOT be
> on the line. [Hence no biggest natural number X on the line. Hence there
> are infinitely many natural numbers on the line. qed.]
>
Btw, note that we accept [the notion of] actual infinity here. We are
talking about an ACTUAL object here, if we are talking about "the nn line".
Try to get that straight: "potential infinity" does not refer to _objects_,
but to _processes_ (or comparable ... whatever). An object, say a set,
can't be "potentially infinite". Either it is finite or (actual) infinite.
Same with the number line.
>>
>> If we only consider the (finite) ordinal numbers proceeding the (infinite)
>> ordinal w, we have
>>
>> 0 1 2 3 ...
>> |----|----|----|- ...
>>
>> i.e. we have infinitely many (finite) numbers.
>>
> And we agree, there are infinitely many numbers on the natural number
> line.
>
See?! :-)
>
> Normally, such contradictions ...
>
Which contradiction[s]? You didn't present a single one so far.
Actually, my impression is that you mix up "contradictions" (referring to
antinomies in a mathematical theory) with "counter-intuitive mathematical
facts".
G. Frege
Actually, my impression is that he mix up (i.e. mis-takes) "contradictions"
G. Frege
>>
>> There are infinitely many natural numbers, but each and any is
>> finite.
>
> That is the current claim. If true, it should not lead to true
> paradoxes ...
>
Right. It DOESN'T.
>
> only apparent ones.
>
Indeed! This is exactly my point. I just wrote (in some other post):
"Actually, my impression is that you mix up 'contradictions'
(referring to antinomies in a mathematical theory) with
'counter-intuitive mathematical facts'."
With other words, imho you are mixing up "true paradoxes" with "apparent
ones".
>>
>> I'll agree that this MIGHT _sound_ strange. Maybe the following
>> _formulation_ is easier to digest:
>>
>> Each and any natural number is finite.
>>
>> But the set of _all_ natural numbers is not bound.
>> (I.e. it does not have an upper boundary.) With other
>> words, there is no biggest natural number. (I guess
>> that's immediately clear, no? If W would be this number,
>> W+1 would be another natural number, but bigger. Which
>> is an absurdity.)
>
> Being unbound gives you EITHER potential or actual infinity.
>
Nonsense.
We have to make a decision BEFORE that. Maybe you know that "transfinite
set theory" (i.e. Cantor) accepted the [notion of] "actual infinity". Hence
we are always talking about "actually infinite" objects here (in the
context of your "problem"). Hence in modern axiomatic set theory and/or
mathematics we do not have to use the attribute "actual" any more, we just
stick to the term "infinite".
Hope this helps.
G. Frege
<David...@alumdotmit.edu> wrote:
>>
>> -- which must be true if there are indeed infinitely many
>> numbers, all of which are finite
>>
> How does this imply the preceding?
>
I guess, the "argument" is something like
Hey, that _sounds_ right, hence...
:-) (Note the "must" in his claim!)
>>
>> However, adding that
>> all natural numbers are finite (in magnitude) DOES tell us the answer,
>> because if we try to say there are infinitely many numbers, we start
>> creating paradoxes, i.e., contradictions (I know of three, but I am sure
>> there are many more).
>>
> What are the three?
>
Actually, my impression is that he is mixing up "contradictions"
(referring to antinomies in a mathematical theory) with "counter-
intuitive mathematical facts".
Of course, if he's not able to understand and appreciate that difference
he's certainly on the edge of crankhood.
Phil
> On Mon, 05 Mar 2007 22:14:19 GMT, Phil <toob-h...@sbcglobal.net>
> said:
>
>>...
>>Anyway, briefly, the argument was that if there are infinitely many
>>numbers, then there must exist at least one number x (actually,
>>infinitely many numbers), such that there are infinitely many numbers
>>preceeding x on the natural number line.
>
>
> I'm afraid I don't see an argument there, only an assertion, moreover
> one that is easily show to be false in set theory.
That's okay, there is no reason why you should instantly see everything
the way I do! However, I want to point out that I often see people here
"prove" that the line paradox "is false" by quoting the arguments which
show that there are indeed infinitely many numbers, all finite. You
cannot prove that a criticism of two arguments is false by quoting those
arguments, in effect saying, "they are too true!" Now, maybe you weren't
planning on doing that, but it kind of looks like it, and almost
everyone else does try to do that.
More exactly, it is
> easy to prove in set theory that (1) there are infinitely many natural
> numbers, and (2) for any natural number n, there are only finitely many
> natural numbers preceding n.
See what I mean?
Is it your claim that there is a flaw in
> the usual proofs of these theorems of set theory?
No.
If so, what is the
> flaw?
>
The flaw is not in either of the arguments themselves, which are fine as
far as I can see. The problem is that they rest on valid, but DIFFERENT,
premises. An argument says that since n is finite, and (n+1) is finite,
that all n's are finite. Fine, although we should be consciously aware
that this implicitly assumes potential infinity, not actual infinity. I
could say that since 1 is preceded by finitely many numbers (namely
zero), and that whenever n is preceded by finitely many numbers, so is
(n+1), that this proves that there are finitely many numbers preceding
any n, just as you say above.
But this is not the standard claim, because this means that there are
finitely many numbers. Proof: given that the natural numbers are
well-ordered, we can easily form a set from 1 to any number n. If the
set of numbers for ANY number n contains finitely many numbers, then it
is impossible to form a set with infinitely many members from the set of
natural numbers. This can only occur if the set of natural numbers does
not contain infinitely many numbers.
You just used potential infinity to define the quantity of numbers in
the set of natural numbers, creating finitely many numbers, rather than
infinitely many numbers (although the numbers are "bounded" only by
potential infinity, and not a specific -- meaning specifiable relative
to zero -- finite number). I think this is a valid option, one that when
combined with the argument showing that all numbers are finite, produces
a set of characteristics that do NOT lead to contradictions and paradoxes.
Nice, but that is not what mathematicians claim. They claim that there
are infinitely many natural numbers, and in that case, we can remove any
finite number of numbers whatsoever -- call these removed numbers set A
-- and still be left with infinitely many numbers remaining, call it set
B. Okay, take set B, and beginning with position 1, create a new natural
number line. Now add set A to the end of the line. Why not? The entire
set was used to create "the" natural number line, what "mystic force" is
going to prevent us from simply repeating that act, placing set B on the
line first, then placing set A. Positionally speaking, we will do
nothing, meaning occupy no position, that was not done when creating the
original natural number line (notice that we are assuming that we have
access to actual infinity when creating any of these lines). Every
number from set A will then be PRECEDED by the infinitely many numbers
from set B. And yet, according to the current beliefs -- based on the
combination of two arguments, one argument using potential infinity, and
the other argument using actual infinity -- the numbers from set A, all
of which are proceeded by infinitely many numbers, are all FINITE, which
means they do NOT equal the number of numbers proceeding them (minus 1).
I should mention that I first formed this paradox using distance from
zero or one. It was George Dance who changed it (for the better) to
using the number of numbers preceding n.
Want another one? Since all numbers are finite, all numbers have a
finite number of digits. Since the maximum possible number of numbers
for a given number of digits n is 10^n, or (10 * 10 * 10 * ... * 10),
i.e., 10 multiplied by itself n number of times, a FINITE number of
times, the maximum number of numbers in the set of natural numbers is
FINITE, since a finite number multiplied by itself a finite number of
times can never be infinite. Want a different version? If n and 10^n are
both finite, then (n+1) and 10^(n+1) are also both finite, for any
finite n whatsoever. Otherwise, we have 10^n = x = finite, but 10^(n+1)
= 10x = infinite for some FINITE number n. Do you really want to claim
that a finite number x exists such that 10x is infinite?
Notice that if we either use actual infinity both BOTH the number of
natural numbers AND the largest value of a number, that both of the
above paradoxes vanish. Ditto if we use potential infinity both the
number of numbers, and the maximum value of a number. It is ONLY when we
"mix and match" potential and actual infinity that we get paradoxes.
Want another one? Take the infinite sequence (1/2 + 1/4 + 1/8 + ...
1/oo). And yes, I realize that 1/oo does not refer to a specific number.
Since every natural number is finite, every one of the values in this
sequence is finite (1/finite = finite). However, since there are
infinitely many of them, we can form a parallel sequence, (1/z + 1/z +
... 1/z), using exactly the same number of elements, but with 1/z being
an infinitesimal, specifically the infinitesimal such that this sequence
adds up to exactly 1, the same as the sequence of inverted natural
numbers (yes, this uses nonstandard analysis, but I am not using it to
redefine the natural numbers, or the natural number line, so get over
it). So, we have two infinite sums, with exactly the same number of
elements, in which EVERY MEMBER of the first sequence is infinitely
larger than EVERY MEMBER of the second sequence, by the definitions of
finite and infinitesimal, and yet they add up to exactly the same thing.
Once again, this paradox vanishes if we do NOT "mix and match" potential
and actual infinity. So, Chris, what do you think? Do you believe that,
100 years from now, mathematicians will say that we can, or cannot, mix
and match potential and actual infinity when combining arguments? I'll
tell you what, why don't you test this yourself, by seeing if you can
come up with your own paradoxes? If everything is just fine now, then
the answer will be no.
Phil
George Dance
> On 5 Mar 2007 18:37:50 -0800, "George Dance" <georgedanc...@yahoo.ca>
> wrote:
>
>
>
> >> But what can I do, George?
>
> > Taking your meds might help.
>
> Can't stand the truth? :-)
>
I have no problem with the truth. The truth is, you sound like a
silly troll who's off his meds.
george
> george wrote:
> > Oh, shut up.
> > Purported proofs that are not actually proofs
> > ARE ALWAYS EASY to refute. ALWAYS.
> > BY DEFINITION.
On Mar 5, 5:39 pm, Phil <toob-head...@sbcglobal.net> wrote:
> Now wait a second. Proofs that contain deeply hidden errors are NOT
> always easy to disprove.
True, but if the error is "deeply" hidden, just WHAT is it deeply
hidden
UNDER? The answer is, a large long number of LINES of proof, THAT'S
what.
THE LENGTH of the proof is what makes the hiding deep.
> That is so obvious that I think we're talking
> about two different things.
Of course we are. I'm talking about proof.
You're talking out your ass.
> I'm saying that when offering an analysis of
> some current belief,
NOBODY is stating any BELIEFS around here, except MAYBE YOU!
Since the name of the group is sci.logic, THE REST of us are all
stating
PROOFS! And whether a proof is or isn't valid IS ALWAYS EASY to
check!
> that simply repeating the current beliefs is not a
> valid refutation of that analysis,
Obviously, the issue is NEVER whether the analysis has been
"reftued", BUT RATHER whether the originally stated belief
was vs. wasn't provable or disprovable.
> > Proof-predicates are PRIMITIVE recursive.
> > The ONLY thing that make a proof HARD is to make it LONG.
> > If the proof is SHORT then checking it IS *ALWAYS* easy.
>
> If people at some time use assumptions/premises that contain errors,
There IS NO SUCH THING as an assumption or premise that
"contains errors"! Your premises are either jointly INCONSISTENT
or they are consistent! If they are inconsistent (and you are lower
than
higher-order logic) then THAT MUST be provable! So THE ONLY valid
objection to them is a proof of their inconsistency! All other
"analyses" of
why they are flawed REALLY CAN just be dismissed by repeating the
original belief.
G. Frege
wrote:
>>
>> More exactly, it is easy to prove in set theory that (1) there are
>> infinitely many natural numbers, and (2) for any natural number n,
>> there are only finitely many natural numbers preceding n.
>>
> See what I mean?
>
Huh?
Look, man, in set theory we can show the existence of a certain set IN
(which we usually call the set of natural numbers), and we can define the
notion of an /infinite set/. Then we can PROVE (with full rogor)
(a) infinite(IN)
(b) An e IN : ~infinite(n)
and
(c) An e IN : ~infinite({m e IN : m < n}).
If you are interested in the proofs you might consult Halmos' book "Naive
Set Theory" for example.
In words, we can PROVE that
(a) The set of all natural numbers is infinite.
(b) All natural numbers are finite.
(c) For any natural number n, the set of all natural
numbers preceding n is finite.
>>
>> Is it your claim that there is a flaw in the usual proofs of these
>> theorems of set theory?
>>
> No.
>
But then, what's your problem? :-o
Or with other words:
>>
>> ...what is the flaw?
>>
> The flaw is not in either of the arguments themselves, which are fine as
> far as I can see.
>
Ah?
>
> The problem is that they rest on valid, but DIFFERENT, premises.
>
Huh?
Note that ALL theorems of (axiomatic) set theory rests just on the _axioms_
of set theory (and the usual axioms and rules of predicate logic).
>
> An argument says that since n is finite, and (n+1) is finite,
> that all n's are finite.
>
Well, that's just an (informal) application of the induction principle. But
since the induction principle can be proved in set theory, there's nothing
wrong with it.
>
> Fine, although we should be consciously aware that this implicitly assumes
> potential infinity, not actual infinity.
>
Nonsense. (Sorry, I mean: Your claim is wrong.)
>
> I could say that since 1 is preceded by finitely many numbers (namely
> zero), and that whenever n is preceded by finitely many numbers, so is
> (n+1), that this proves that there are finitely many numbers preceding
> any n, just as you say above.
>
Right. Your argument is ok.
>
> But this is not the standard claim, because this means that there are
> finitely many numbers.
>
Huh? Non sequitur. Funny thing with you. First you argue in a modest and
coherent way. But suddenly you pull out a rabbit from some hat... :-)
Again: No, this does not "mean that there are [only] finitely many
[natural] numbers."
Why should it? Where's the ARGUMENT for such a claim. (Hint: There is no
valid argument for this claim, since it is simply wrong.)
>
> Proof: given that the natural numbers are well-ordered, we can easily
> form a set from 1 to any number n.
>
Ok. Let
A_n =df {m e IN\{0} : m <= n} ,
where n is a natural number.
>
> If the set of numbers for ANY number n contains finitely many numbers,
> then it is impossible to form a set with infinitely many members from
> the set of natural numbers.
>
Huh??? This is a (WRONG) claim. Not an argument (i.e. a mathematical
proof).
Look, it's indeed true that each and any set A_n is finite (with n e IN).
But that does not mean that IN itself is finite.
Moreover, if you consider the (so called) UNION of _all_ sets A_n (with n e
IN) then you get an _infinite_ set, namely IN:
U A_n = IN.
n e IN
Again, this can easily be PROVED with full rigor. (But I'm simply to lazy
to write down the proof.)
>
> This can only occur if the set of natural numbers does not contain
> infinitely many numbers.
>
???
Look, Phil, there's something you didn't get so far it seems: A
mathematical proof doesn't just consist of some claims. [*sigh*]
>
> You just used potential infinity to define the quantity of numbers in
> the set of natural numbers,
>
Huh??? Where the fuck do get all this nonsense from? :-o
>
> ...creating finitely many numbers, rather than
> infinitely many numbers [...]. I think this is a valid option, one that when
> combined with the argument showing that all numbers are finite, produces
> a set of characteristics that do NOT lead to contradictions and paradoxes.
>
Look, man, this is a "intuitionistic" point of view, but NOT the standard
view of classical mathematics, i.e. standard axiomatic set theory.
>
> Nice, but that is not what mathematicians claim.
>
Exactly!
>
> They claim that there are infinitely many natural numbers,
>
Exactly.
Again, this can be PROVED (see comments above). With other words, this is
not just a CLAIM but a THEOREM of set theory.
>
> and in that case, we can remove any
> finite number of numbers whatsoever -- call these removed numbers set A
> -- and still be left with infinitely many numbers remaining, call it set
> B.
>
Right. Ok.
>
> Okay, take set B, and beginning with position 1, [consider] a new natural
> number line. Now add set A to the end of the line. Why not?
>
Because THERE IS NO END of the line. (Hence you CAN'T do that.)
>
> The entire
> set was used to create "the" natural number line, what "mystic force" is
> going to prevent us from simply repeating that act, placing set B on the
> line first, then placing set A.
>
The "mystic force" is called LOGIC and CONSISTENCE.
If there IS NO end you hardly can add something to this (non-existent) end.
Again, it's clear from what you write here that you mis-take the nn-line
with a "line" consisting of the natural numbers and infinite ordinals. i.e.
such a "line":
0 1 2 3 ... w w+1 w+2 ...
|----|----|----|- ... -|----|----|-- ...
Here we might take away some finite part of the "line" at its beginning.
This way we would get:
0 1 2 3 .. a
|----|----|----|- .. -|
a+1 a+2 a+3 a+4 ... w w+1 w+2 ...
|----|----|----|- ... -|----|----|-- ...
And, of course, we might _replace_ the finite many (infinite) ordinal
numbers w, w+1, .. w+a with 0, 1, 2, .., a. This way we would get:
a+1 a+2 a+3 a+4 ... 0 1 2 .. a w+a+1 ...
|----|----|----|- ... -|----|----|- .. -|----|--- ...
>
> Positionally speaking, we will do
> nothing, meaning occupy no position, that was not done when [considering] the
> original natural number line
> ~~~~~~~~~~~~~~~~~~~
Nope. Not the original nn line, but a "line" consisting of all natural
numbers (i.e. finite ordinals) + infinite ordinals.
>
> [...] Every
> number from set A will then be PRECEDED by the infinitely many numbers
> from set B.
>
Right. See the final diagram above.
>
> And yet, according to the current beliefs [...] the numbers from set A, all
> of which are proceeded by infinitely many numbers, are all FINITE, which
> means they do NOT equal the number of numbers proceeding them (minus 1).
>
Right. That's indeed the case (here).
>
> I should mention that I first formed this paradox using distance from
> zero or one. It was George Dance who changed it (for the better) to
> using the number of numbers preceding n.
>
Fine. But PLEASE stick to the _nn line_, ok?
>
> Want another one?
>
Another what?
>
> Since all numbers are finite, all numbers have a finite number of digits.
>
Right.
>
> Since the maximum possible number of numbers
> for a given number of digits n is 10^n, or (10 * 10 * 10 * ... * 10),
> i.e., 10 multiplied by itself n number of times, a FINITE number of
> times, the maximum number of numbers in the set of natural numbers is
> FINITE,
>
Huh? Again another rabbit out of the hat! :-)
>
> since a finite number multiplied by itself a finite number of
> times can never be infinite.
>
So what? How does this FACT prove that _the set of all natural numbers_ is
finite? :-o
>
> Want a different version?
>
Actually a CORRECT one would suffice.
>
> If n and 10^n are both finite, then (n+1) and 10^(n+1) are also both finite,
> for any finite n whatsoever.
>
Right!
>
> Otherwise, we have 10^n = x = finite, but 10^(n+1)
> = 10x = infinite for some FINITE number n.
>
Huh? Where the fuck do get all this nonsense from?! :-o
>
> Do you really want to claim
> that a finite number x exists such that 10x is infinite?
>
No. WE do not claim that. It seems that you miss the point that YOU are the
one who is claiming that nonsense.
>
> Want another one?
>
PLEASE, PLEASE, A CORRECT ONE!
>
> Take the infinite sequence (1/2 + 1/4 + 1/8 + ...
> 1/oo).
> ~~~~
>
There is no such sequence. Sorry.
>
> And yes, I realize that 1/oo does not refer to a specific number.
>
SO WHY -the fuck- do you WRITE IT, in the first place?
>
> Since every natural number is finite, every one of the values in this
> sequence is finite (1/finite = finite). However, since there are
> infinitely many of them, we can form a parallel sequence, (1/z + 1/z +
> ... 1/z), using exactly the same number of elements, but with 1/z being
> an infinitesimal, ...
>
There are no infinitesimals in standard Analysis. And I doubt that you are
familiar enough with Nonstandard-Analysis to formulate a CORRECT argument.
Sorry.
>
> Once again, this paradox[es would] vanishes...
>
..if you would stop talking nonsense. Try it!
Aatu Koskensilta
> Well, that's just an (informal) application of the induction principle. But
> since the induction principle can be proved in set theory, there's nothing
> wrong with it.
Would there be something wrong in it if it wasn't provable in set theory?
(Your diagrams are cute).
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
G. Frege
<aatu.kos...@xortec.fi> wrote:
>>
>> Well, that's just an (informal) application of the induction principle. But
>> since the induction principle can be proved in set theory, there's nothing
>> wrong with it.
>>
> Would there be something wrong in it if it wasn't provable in set theory?
>
Well... a moot question, since it IS provable in set theory. ;-)
David Marcus
> Chris Menzel wrote:
>
> > I'm afraid I don't see an argument there, only an assertion, moreover
> > one that is easily show to be false in set theory.
>
> That's okay, there is no reason why you should instantly see everything
> the way I do! However, I want to point out that I often see people here
> "prove" that the line paradox "is false" by quoting the arguments which
> show that there are indeed infinitely many numbers, all finite. You
> cannot prove that a criticism of two arguments is false by quoting those
> arguments, in effect saying, "they are too true!" Now, maybe you weren't
> planning on doing that, but it kind of looks like it, and almost
> everyone else does try to do that.
> > More exactly, it is
> > easy to prove in set theory that (1) there are infinitely many natural
> > numbers, and (2) for any natural number n, there are only finitely many
> > natural numbers preceding n.
>
> See what I mean?
I see people doing two things:
1. Giving a proof that your statement is false.
2. Pointing out where your "proof" of your statement has a gap.
If your "proof" isn't a proof, then the paradox disappears.
> > Is it your claim that there is a flaw in
> > the usual proofs of these theorems of set theory?
>
> No.
>
> > If so, what is the
> > flaw?
>
> The flaw is not in either of the arguments themselves, which are fine as
> far as I can see.
What two arguments? Yours and the standard one?
> The problem is that they rest on valid, but DIFFERENT,
> premises. An argument says that since n is finite, and (n+1) is finite,
> that all n's are finite. Fine, although we should be consciously aware
> that this implicitly assumes potential infinity, not actual infinity. I
> could say that since 1 is preceded by finitely many numbers (namely
> zero), and that whenever n is preceded by finitely many numbers, so is
> (n+1), that this proves that there are finitely many numbers preceding
> any n, just as you say above.
>
> But this is not the standard claim, because this means that there are
> finitely many numbers. Proof: given that the natural numbers are
> well-ordered, we can easily form a set from 1 to any number n. If the
> set of numbers for ANY number n contains finitely many numbers, then it
> is impossible to form a set with infinitely many members from the set of
> natural numbers.
Here is your gap. You haven't shown that
(1) The set of numbers for ANY number n contains finitely many numbers
implies
(2) It is impossible to form a set with infinitely many members from the
set of natural numbers.
The gap in this argument is you aren't clear on what "FINITE" means. The
usual definition depends on the order of the numbers. However, you are
changing the order. So, you need to be clear what definition you are
using.
> Want another one? Since all numbers are finite, all numbers have a
> finite number of digits. Since the maximum possible number of numbers
> for a given number of digits n is 10^n, or (10 * 10 * 10 * ... * 10),
> i.e., 10 multiplied by itself n number of times, a FINITE number of
> times, the maximum number of numbers in the set of natural numbers is
> FINITE, since a finite number multiplied by itself a finite number of
> times can never be infinite.
The gap here is that just because the number of numbers you can express
with n digits is finite for each n, doesn't imply that the total number
of numbers is finite.
> Want a different version? If n and 10^n are
> both finite, then (n+1) and 10^(n+1) are also both finite, for any
> finite n whatsoever. Otherwise, we have 10^n = x = finite, but 10^(n+1)
> = 10x = infinite for some FINITE number n.
The gap here is that you haven't shown that 10x is infinite for some
finite n.
> Do you really want to claim
> that a finite number x exists such that 10x is infinite?
--
David Marcus
Aatu Koskensilta
The induction principle is provable in set theory. But what is the point in
noting that there and implying that that is why "there's nothing wrong with"
the induction principle (or the specific application?)?
G. Frege
<aatu.kos...@xortec.fi> wrote:
>>
>> Well... a moot question, since it IS provable in set theory. ;-)
>>
> The induction principle is provable in set theory. But what is the point in
> noting that there and implying that that is why "there's nothing wrong with"
> the induction principle (or the specific application?)?
>
Ah, now I see the relevance of your question. Well, the point is (was) to
make clear that the induction principle is not just some "claim", but a
theorem of set theory (i.e. something which can be _proved_ with full
rigor).
You see, somehow, good old Phil is mixing up "claims" with "theorems", it
seems.
G. Frege
<David...@alumdotmit.edu> wrote:
>
> If your "proof" isn't a proof, then the paradox disappears.
>
> Here is your gap. [...].
> The gap in this argument is [...].
> The gap here is that [...].
> The gap here is that [...].
>
Paradox vanished!
Chris Menzel
> Chris Menzel wrote:
>
>> On Mon, 05 Mar 2007 22:14:19 GMT, Phil <toob-h...@sbcglobal.net>
>> said:
>>>... Anyway, briefly, the argument was that if there are infinitely
>>>many numbers, then there must exist at least one number x (actually,
>>>infinitely many numbers), such that there are infinitely many numbers
>>>preceeding x on the natural number line.
>>
>> I'm afraid I don't see an argument there, only an assertion, moreover
>> one that is easily show to be false in set theory.
>
> That's okay, there is no reason why you should instantly see
> everything the way I do! However, I want to point out that I often see
> people here "prove" that the line paradox "is false" by quoting the
> arguments which show that there are indeed infinitely many numbers,
> all finite. You cannot prove that a criticism of two arguments is
> false by quoting those arguments,
Granted, *if* all one does is quote them.
> ...in effect saying, "they are too true!" Now, maybe you weren't
> planning on doing that, but it kind of looks like it, and almost
> everyone else does try to do that.
>
>> More exactly, it is easy to prove in set theory that (1) there are
>> infinitely many natural numbers, and (2) for any natural number n,
>> there are only finitely many natural numbers preceding n.
>
> See what I mean?
No I do not, since I go on and ask:
>> Is it your claim that there is a flaw in the usual proofs of these
>> theorems of set theory?
>
> No.
>
> If so, what is the flaw?
> The flaw is not in either of the arguments themselves, which are fine
> as far as I can see.
But then I am very confused, because above you said you were offering
*criticisms* of the arguments. Generally speaking there are two ground
on which you can criticize an argument: (1) The argument is not
logically valid, or (2) it is logically valid but at least one of its
premises is false, i.e., the argument is valid but not sound. But if,
as you say, the arguments are "fine", by which I can only imagine you
mean the argument is sound, then it follows that you accept their
conclusions. For you cannot rationally accept the premises of a sound
argument without accepting its conclusion. So I do not see anything
like a *criticism* in your remarks.
But perhaps what you are saying is that we get a *paradox* precisely
because these arguments are apparently sound, but that you've got
*other* apparently sound arguments that lead to conclusions that
contradict the conclusions of the arguments above. That would indeed be
a paradox.
> The problem is that they rest on valid, but DIFFERENT, premises.
How can they rest on DIFFERENT premises? They rest on exactly the
premises that they rest on.
> An argument says that since n is finite, and (n+1) is finite, that all
> n's are finite.
Not quite. The argument is that, because (i) 0 is finite and (ii) for
any natural number n, if n is finite, then n+1 is finite, it follows by
mathematical induction that all natural numbers are finite.
> Fine, although we should be consciously aware that this implicitly
> assumes potential infinity, not actual infinity.
There is no mention of infinity there at all. Only finitude. Moreover,
there is no rigorous notion (in standard set theory) of potential
infinity. There is just the one notion "infinite" (though granted it
requires the axiom of choice to prove that two relatively intuitive
notions of the infinite are equivalent).
> I could say that since 1 is preceded by finitely many numbers (namely
> zero), and that whenever n is preceded by finitely many numbers, so is
> (n+1), that this proves that there are finitely many numbers preceding
> any n, just as you say above.
>
> But this is not the standard claim, because this means that there are
> finitely many numbers. Proof: given that the natural numbers are
> well-ordered, we can easily form a set from 1 to any number n.
True enough.
> If the set of numbers for ANY number n contains finitely many numbers,
I guess you mean: If the set of numbers less than ANY given number
contains finitely many numbers...
> then it is impossible to form a set with infinitely many members from
> the set of natural numbers.
It puzzles me that you don't see that there is a very large gap here.
On the face of it, it simply doesn't follow. You need some such premise
as that the set of natural numbers can be put into one-to-one
correspondence with the numbers less than some natural number. But
there is a very easy proof that this is impossible. So what is your
argument from:
(1) The set of numbers less than ANY given number contains finitely many
numbers
to
(2) Every set of natural numbers is finite.
I think if you can provide this argument you will be able to make your
case quite convincingly. But you do need to you standard notions of
finite and infinite as found in any elementary set theory text or else
you are talking about concepts no one else is talking about (in this
context). They might be interesting concepts, but you should use
different words for them so it is clear that you are talking about
something else.
> You just used potential infinity to define the quantity of numbers in
> the set of natural numbers,
I really didn't. I used the standard notion found in any elementary set
theory text.
> Nice, but that is not what mathematicians claim. They claim that there
> are infinitely many natural numbers, and in that case, we can remove
> any finite number of numbers whatsoever -- call these removed numbers
> set A -- and still be left with infinitely many numbers remaining,
> call it set B.
That is correct, that is a simple theorem concerning infinite sets.
> Okay, take set B, and beginning with position 1,
I guess you mean the smallest natural number in B.
> create a new natural number line.
I'm not sure what "create" amounts to here, but I think it's pretty
clear that what you have in mind is the natural ordering of B determined
by the usual less-than relation on the natural numbers. Ok. I'm with
you.
> Now add set A to the end of the line.
Well, you haven't clearly defined what you mean by "add A to the end of
the line", but there is such a notion available in the theory of ordinal
numbers -- from a quick look at G Frege's replies to you I think he's
already mentioned that. In this sense of "adding to the end", the
result will be a well-ordered set of order type omega + n, where n is
the cardinality of A.
> Why not? The entire set was used to create "the" natural number line,
> what "mystic force" is going to prevent us from simply repeating that
> act, placing set B on the line first, then placing set A.
Nothing at all. This is commonly done in set theory in the study of
ordinals and well-orderings.
> Positionally speaking, we will do nothing, meaning occupy no position,
> that was not done when creating the original natural number line
> (notice that we are assuming that we have access to actual infinity
> when creating any of these lines).
Couldn't follow this.
> Every number from set A will then be PRECEDED by the infinitely many
> numbers from set B.
Yes, in the new ordering, that is quite correct.
> And yet, according to the current beliefs -- based on the combination
> of two arguments, one argument using potential infinity, and the other
> argument using actual infinity -- the numbers from set A, all of which
> are proceeded by infinitely many numbers, are all FINITE, which means
> they do NOT equal the number of numbers proceeding them (minus 1).
That is also correct, but all you've done is change the subject. You
are no longer talking about the set of natural numbers in their natural
ordering; you are talking about that set with a *different* ordering,
where "precedes" is no longer synonymous with "less than".
> I should mention that I first formed this paradox
Note there is no paradox at all, unless you forget that "precedes" in
the new ordering no longer coincides with "less than".
> Want another one?
That's enough for now. :-) Must get back to work, but would be happy to
look at other arguments later.
G. Frege
>
> [F.] this looks useful and workable; I'll get back to it later!
>
Fine.
Here's another line of thought which might be helpful, I hope.
<begin quote>
---------------------------------------------
>>
>> Unless there is an infinite number the number of [natural] numbers
>> [...] cannot be infinite. (W. Mückenheim)
>>
I have pondered about that question [a common theme among mathematical
cranks] some time. How about the following argument?
Assume (for the sake of the argument) that in the set of _all_ natural
numbers, denoted by IN*, there were infinite natural numbers (though I
don't have the slightest idea how such numbers would look like). Now
let's construct a set IN the following way:
IN := {n e IN* | n is a finite number}.
Then IN is the set of all finite natural numbers. It MUST be, since it
is "constructed" that way. You might think of it the following way too:
IN := IN* - {n e IN* | n is infinite}
i.e. IN is the set of _all_ natural numbers /minus/ (set theoretic minus)
_all_ the infinite natural numbers. Hence IN is the reminder: the set
of all natural numbers that are not infinite, hence finite.
Now it's easy to see that IN (in contrast to WM's claim) is still
_infinite_, though only containing _finite_ natural numbers. This
shows/proves that his claim ("argument") is false.
---------------------------------------------
<end quote>
G. Frege
>
> Here's another line of thought which might be helpful, I hope.
>
> <begin quote>
>---------------------------------------------
>
>>>
>>> Unless there is an infinite number the number of [natural] numbers
>>> [...] cannot be infinite. (W. Mückenheim)
>>>
> I have pondered about that question [a common theme among mathematical
> cranks] some time. How about the following argument?
>
> Assume (for the sake of the argument) that in the set of _all_ natural
> numbers, denoted by IN*, there were infinite natural numbers (though I
> don't have the slightest idea how such numbers would look like). Now
> let's construct a set IN the following way:
>
> IN := {n e IN* | n is a finite number}.
>
> Then IN is the set of all finite natural numbers. It MUST be, since it
> is "constructed" that way. You might think of it the following way too:
>
> IN := IN* - {n e IN* | n is infinite}
>
> i.e. IN is the set of _all_ natural numbers /minus/ (set theoretic minus)
> _all_ the infinite natural numbers. Hence IN is the reminder: the set
> of all natural numbers that are not infinite, hence finite.
>
> Now it's easy to see that IN (in contrast to WM's claim) is still
> _infinite_, though only containing _finite_ natural numbers. This
> shows/proves that his claim ("argument") is false.
>
>---------------------------------------------
><end quote>
>
A (slight) variant:
<begin quote>
---------------------------------------------
> So you can have an infinite set N but no infinite member of it.
Right.
I have pondered about that question some time (motivated by the
inability of many cranks to comprehend that "state of affairs").
Many cranks (Mückenheim for example) claim that (from a logical point
of view) it's not possible for a set of natural numbers to be infinite
if it's only containing finite numbers. Well, let's see...
Assume (for the sake of the argument) that in the set of _all_ natural
numbers, denoted by IN*, there were infinite natural numbers (though I
don't have the slightest idea how such numbers would look like). Now
let's construct [or rather define] a set IN the following way:
IN = {n e IN* | n is a finite number}.
Then IN is the set of all finite natural numbers. It MUST be, since it
is "constructed" that way. You might think of it the following way too:
IN = IN* - {n e IN* | n is infinite}
i.e. IN is the set of _all_ natural numbers /minus/ (set theoretic minus)
_all_ the infinite natural numbers that are in IN*. Hence IN is the
reminder: the set of all natural numbers that are not infinite, hence
finite.
Now the crucial question just is: Is IN finite then? C a n it be finite?
If the cranks were right, it _should_ be! After all it consists of exactly
all the natural numbers that are finite.
You might try to find the answer to this question yourself.
---------------------------------------------
<end quote>
Hope this helps.
Phil
[snip]
>>>
>>>I'm afraid I don't see an argument there, only an assertion, moreover
>>>one that is easily show to be false in set theory.
>>
>>That's okay, there is no reason why you should instantly see
>>everything the way I do! However, I want to point out that I often see
>>people here "prove" that the line paradox "is false" by quoting the
>>arguments which show that there are indeed infinitely many numbers,
>>all finite. You cannot prove that a criticism of two arguments is
>>false by quoting those arguments,
>
>
> Granted, *if* all one does is quote them.
>
>
>>...in effect saying, "they are too true!" Now, maybe you weren't
>>planning on doing that, but it kind of looks like it, and almost
>>everyone else does try to do that.
>>
>>
>>>More exactly, it is easy to prove in set theory that (1) there are
>>>infinitely many natural numbers, and (2) for any natural number n,
>>>there are only finitely many natural numbers preceding n.
>>
>>See what I mean?
>
>
> No I do not, since I go on and ask:
You said that it is easy to prove (1) and (2). I said that there are
flaws in the current beliefs about the natural numbers, beliefs which
are BASED on (1) and (2). You CANNOT "prove" that there are no problems
with the COMBINATION of (1) and (2), by RESTATING that (1) and (2), in
and of themselves, are valid. To simply restate that mathematicians
believe that the conclusions (1) and (2) are correct (which I agree
with), in no way "proves" that , and that COMBINING (1) and (2) in the
natural numbers -- and since we need both (1) and (2) to define the
natural numbers, we do indeed have to combine them -- we do not cause
problems, paradoxes, contradictions. Specifically, the fact that (1) and
(2) are valid does NOT prove that the premises behind (1) and (2) --
including any hidden or unspoken premises -- are compatible. Now, do we
at least agree on a POTENTIAL source of problems with the natural
numbers, even granted that (1) and (2) are valid? You may BELIEVE that
mathematicians have found every single premise behind (1) and (2), but
surely, your scientific training has told you that we humans can NEVER
assume that we have, indeed, found every premise behind some logical
structure. In other words, surely you agree that your belief that we
know ALL the premises behind (1) and (2), and that they are completely
compatible, is an opinion, and not some god-given absolute truth, yes?
>
>
>>>Is it your claim that there is a flaw in the usual proofs of these
>>>theorems of set theory?
>>
>>No.
>>
>>If so, what is the flaw?
>
>
>>The flaw is not in either of the arguments themselves, which are fine
>>as far as I can see.
>
>
> But then I am very confused, because above you said you were offering
> *criticisms* of the arguments. Generally speaking there are two ground
> on which you can criticize an argument: (1) The argument is not
> logically valid, or (2) it is logically valid but at least one of its
> premises is false, i.e., the argument is valid but not sound. But if,
> as you say, the arguments are "fine", by which I can only imagine you
> mean the argument is sound, then it follows that you accept their
> conclusions. For you cannot rationally accept the premises of a sound
> argument without accepting its conclusion. So I do not see anything
> like a *criticism* in your remarks.
>
> But perhaps what you are saying is that we get a *paradox* precisely
> because these arguments are apparently sound, but that you've got
> *other* apparently sound arguments that lead to conclusions that
> contradict the conclusions of the arguments above. That would indeed be
> a paradox.
>
>
>>The problem is that they rest on valid, but DIFFERENT, premises.
>
>
> How can they rest on DIFFERENT premises? They rest on exactly the
> premises that they rest on.
Let's try this again. Although each of the two arguments -- and again, I
completely believe in both of them -- do indeed rest on "the premises
they rest on," they rest on different SETS of premises. The set of
premises behind argument 1 is NOT the same as the set of premises behind
argument 2. This is the reason that the paradoxes exist. The paradoxes
are both an indication, and a demonstration, that the two sets of
underlying premises are not the same.
>
>
>>An argument says that since n is finite, and (n+1) is finite, that all
>>n's are finite.
>
>
> Not quite. The argument is that, because (i) 0 is finite and (ii) for
> any natural number n, if n is finite, then n+1 is finite, it follows by
> mathematical induction that all natural numbers are finite.
>
>
>>Fine, although we should be consciously aware that this implicitly
>>assumes potential infinity, not actual infinity.
>
>
> There is no mention of infinity there at all. Only finitude. Moreover,
> there is no rigorous notion (in standard set theory) of potential
> infinity. There is just the one notion "infinite" (though granted it
> requires the axiom of choice to prove that two relatively intuitive
> notions of the infinite are equivalent).
>
>
>>I could say that since 1 is preceded by finitely many numbers (namely
>>zero), and that whenever n is preceded by finitely many numbers, so is
>>(n+1), that this proves that there are finitely many numbers preceding
>>any n, just as you say above.
>>
>>But this is not the standard claim, because this means that there are
>>finitely many numbers. Proof: given that the natural numbers are
>>well-ordered, we can easily form a set from 1 to any number n.
>
>
> True enough.
>
>
>>If the set of numbers for ANY number n contains finitely many numbers,
>
>
> I guess you mean: If the set of numbers less than ANY given number
> contains finitely many numbers...
No, I mean that the finite induction argument above ends with the
conclusion that for ANY n whatsoever, 1 through n is a finite, not an
infinite, number of numbers. Let's go through a couple of these finite
inductions:
(A) 1 = the number of numbers from 1 through 1 inclusive. If n = the
number of numbers from 1 through n inclusive, then (n+1) = the number of
numbers from 1 through (n+1) inclusive. Therefore, every number n = the
number of numbers from 1 through n inclusive.
(B) 1 through 1 is a finite number of numbers. If 1 through n is a
finite number of numbers, then so is 1 through (n+1), otherwise there is
a number n -- since n = the number of numbers from 1 thorugh n inclusive
-- such that n is finite, but (n+1) is infinite. Therefore, the number
of numbers from 1 through n is finite for any number n.
Notice that finite induction does NOT say that for some SPECIFIABLE
number n, the conclusions of finite induction are true. For example,
finite induction is used to prove that every number n in the set of
natural numbers is finite. Fine, but this does NOT mean that for any
number n which we can actually SPECIFY, that n is finite. It means that
for any number n in the ENTIRE SET of natural numbers PERIOD, that n is
finite. Therefore, the results from the finite induction argument (B)
used above, namely that the quantity of numbers from 1 through n is
finite for ANY number n, show that there is no natural number n, in the
entire set of natural numbers, such that 1 through n is infinite. This
means that, according to (B), there is no subset within N that contains
infinitely many numbers.
Note that you cannot say, "Well, something besides finite induction says
that there are infinitely many natural numbers, therefore argument (B)
does not say that no subset of N contains infinitely many numbers,"
because (B) DOES say that. It's just that mathematicians did not use
finite induction -- and the premises that lie behind finite induction --
to determine the number of natural numbers. Instead, they used a
different method, one with different premises. That fact alone should
give you pause, not because it proves that a problem does exist, but
because it proves that a problem might exist.
Given that finite induction says that no number is infinite, and that no
subset of N contains infinitely many numbers (else an n MUST exist such
that 1 through n contains infinitely many numbers), we need to ask why
mathematicians believe that there are infinitely many numbers. I tend to
agree with you that they believe, or perhaps assume, that potential
infinity and actual infinity are one and the same, but we can TEST that
belief by asking whether the characteristics of the natural numbers do
or do not lead to paradoxes.
Let's cut to the chase; either you are going to admit that on a line
with infinitely many numbers, there are subsets/sections of the line
that contain infinitely many numbers, or you're not. If you are going to
claim that SINCE valid arguments exist that state that there are (1)
infinitely many numbers, (2) all of which are finite, and that since (3)
mathematicians are fantastic experts who never overlook conflicting
premises, spoken or unspoken, then (1) and (2) cannot possible be based
on different premises, so the line must not have any n whatsoever, such
that 1 through n contain infinitely many numbers, even though the line
contains infinitely many numbers, then there's really no point in
continuing this discussion. This is the "filibuster logic" I was
referring to, which states that SINCE the current beliefs are perfect,
all paradoxes are false or non-existent. But Damnit, Chris, finite
induction proves that all numbers are finite, and this HAS to be
consistent with the finite induction proof that there are NOT infinitely
many numbers, so why would you ASSUME, or just BELIEVE, that it is also
consistent with a very different conclusion, based on different
premises, that states that there are infinitely many numbers? I'll take
a brief look at what follows, but I think we need to "start from
scratch," or at least, from this point and move on.
>
>
>>then it is impossible to form a set with infinitely many members from
>>the set of natural numbers.
>
>
> It puzzles me that you don't see that there is a very large gap here.
> On the face of it, it simply doesn't follow. You need some such premise
> as that the set of natural numbers can be put into one-to-one
> correspondence with the numbers less than some natural number. But
> there is a very easy proof that this is impossible. So what is your
> argument from:
>
> (1) The set of numbers less than ANY given number contains finitely many
> numbers
>
> to
>
> (2) Every set of natural numbers is finite.
>
> I think if you can provide this argument you will be able to make your
> case quite convincingly. But you do need to you standard notions of
> finite and infinite as found in any elementary set theory text or else
> you are talking about concepts no one else is talking about (in this
> context). They might be interesting concepts, but you should use
> different words for them so it is clear that you are talking about
> something else.
>
>
>>You just used potential infinity to define the quantity of numbers in
>>the set of natural numbers,
>
>
> I really didn't. I used the standard notion found in any elementary set
> theory text.
Your use of a concept depends only on whether you actually use it, not
on whether you are aware of using it or not. If you use the concept of
unbounded, but not infinite, then you are using potential infinity.
David Marcus
> Chris Menzel wrote:
OK up to here. But, next there is a gap in your argument.
> This
> means that, according to (B), there is no subset within N that contains
> infinitely many numbers.
How do you get this from (B)? It doesn't follow.
What principles are you taken as given regarding the natural numbers?
The Peano axioms?
--
David Marcus
Phil
All right, I'll give you one more chance, even though you've convinced
me that you have no interest in being honest in your discussions. All
subsets of 1 through n that do not use all the elements from 1 through n
have fewer elements than the set from 1 through n. If you even THINK
about asking me "How do you know this," I swear I'll spit on you next
time I see you. Well, okay, you can get the set 1 thru n by combining
two sets, the unused elements plus the used elements. I think even you
can take it from there. "How do you know you're not dreaming?" Enough.
Since every subset of 1 thru n contains =< elements, it follows that we
only need to worry about the sets 1 thru n when it comes to maximum
size. Since the set 1 thru n is finite for EVERY n -- and since no
subset of 1 thru n contains more elements than 1 thru n -- there is no
subset of N (the set of all natural numbers) that contains infinitely
many numbers.
As a side note, we have to be aware of and consistent about the
abilities and limitations of finite induction. If we cannot take the
conclusions from finite induction about the number of natural numbers
seriously, why should we take its conclusions about whether all natural
numbers are finite seriously? Is it to be taken seriously at all? I say
yes, but we have to understand its premises, and its "point of view," or
characteristics in general, or we run the risk of making errors when
combining the conclusions of finite induction with the conclusions given
by other techniques.
>
> What principles are you taken as given regarding the natural numbers?
> The Peano axioms?
>
Yes, although finite induction obviously is, not the "weak" point, but
the "fuzzy" point. And of course, its "fuzziness" exists in the area of
infinity. Technically, I see finite induction as perfectly valid given
the assumption of potential infinity. For example, in the Achilles, for
any point x a finite distance from the door, x/2 will also be a finite
distance from the door. Under finite induction, therefore, we can never
leave the room, and this is correct as long as we are limited to
potential infinity in terms of both the rate at which we cross halfway
points (points per second), and the amount of time that we are allowed
to cross those points (less than an infinite amount of time).
As another side note, Aristotle proved that all human knowledge is based
on undefinable, unprovable concepts, mostly our sensory perceptions of
the world, such as volume, heavy, rough, bright, loud, etc. Therefore,
the clearest, most understandable definitions are those that provide the
clearest, most direct link between the concept being defined, and the
basic (more "prior and immediate") concepts that underlie all human
knowledge. In contrast, definitions which are the most IMMUNE to
dialectic assault are those which HIDE their inherent
premises/assumptions as deeply as possible; something which usually
destroys their clarity.
For example, in defining mass, Newton said that it is an object's volume
times its density, and that it is also proportional to an object's
weight. Volume, density, and weight are all basic sensory concepts, ones
which we experience every day, unlike mass, which is rarely directly
experienced. Mach "improved" this definition by giving a method of
measuring mass, by comparing the acceleration given by a spring to an
object with the acceleration given by the spring to a "standard" object.
As Powers pointed out, this only works if the absolute acceleration of
the experiment is zero, meaning it has hidden assumptions, and it is
terrible at making the concept of mass actually clear, but it is VERY
hard to attack in a debate, whereas Mach easily attacked Newton's
definitions as "circular," even though definitions cannot, technically,
be circular (arguments are circular, definitions are merely poor). As a
DEFINITION, Newton's does a very good job of giving a good understanding
of mass, while Mach's is really just an example, and mainly gives an
understanding (not a good one) of the relationship between mass and
force. But modern scientists regard Mach's "definition" as greatly
superior to Newton's, because Mach's is much more difficult to attack in
a debate setting. That's really dumb, but it's also the way it currently is.
Mathematicians also tend to prefer "definitions" that cannot be cut to
shreds in a debate environment, but this is not a good thing as far as
clarity, understanding, and freedom from stupid errors is concerned.
Zeno's Achilles (and Dichotomy) involve points between us and the door,
and walking out of the room, something we do every day. Potential
infinity is EASILY understood here as the kind of infinity that does NOT
allow us to ever leave the room. We can get closer and closer to the
door, but as long as we cross the halfway points at a potentially
infinite rate -- a finite but unbounded rate -- for a potentially
infinite amount of time, we will stay in the room. Actual infinity is
what allows us to cross the ENTIRE SET of halfway points, EVERY halfway
point, between us and the door in a finite period of time. This won't
satisfy mathematicians, and it is really not complete even for an
informal understanding of infinity, but if you want to avoid the kind of
errors that lousy definitions often lead to, it is the kind of
understanding of infinity that you must have (you can have other, more
formal types as well).
Finite induction is fine, but it ALWAYS operates on "this side of the
door," in the potentially infinite realm, and that fact must be
remembered. Whenever we deal with something that involves ACTUAL
infinity, we cannot use finite induction, nor are the conclusions
obtained from finite induction applicable.
Phil
David Marcus
> David Marcus wrote:
> All
> subsets of 1 through n that do not use all the elements from 1 through n
> have fewer elements than the set from 1 through n. If you even THINK
> about asking me "How do you know this," I swear I'll spit on you next
> time I see you. Well, okay, you can get the set 1 thru n by combining
> two sets, the unused elements plus the used elements. I think even you
> can take it from there. "How do you know you're not dreaming?" Enough.
>
> Since every subset of 1 thru n contains =< elements, it follows that we
> only need to worry about the sets 1 thru n when it comes to maximum
> size.
What about the unbounded sets? Why don't we have to worry about them?
> Since the set 1 thru n is finite for EVERY n -- and since no
> subset of 1 thru n contains more elements than 1 thru n -- there is no
> subset of N (the set of all natural numbers) that contains infinitely
> many numbers.
I suppose you mean no bounded set contains infinitely many numbers. But,
you haven't said anything about the unbounded sets.
> > What principles are you taken as given regarding the natural numbers?
> > The Peano axioms?
>
> Yes, although finite induction obviously is, not the "weak" point, but
> the "fuzzy" point. And of course, its "fuzziness" exists in the area of
> infinity. Technically, I see finite induction as perfectly valid given
> the assumption of potential infinity. For example, in the Achilles, for
> any point x a finite distance from the door, x/2 will also be a finite
> distance from the door. Under finite induction, therefore, we can never
> leave the room, and this is correct as long as we are limited to
> potential infinity in terms of both the rate at which we cross halfway
> points (points per second), and the amount of time that we are allowed
> to cross those points (less than an infinite amount of time).
I don't know what you mean by "finite induction". The usual Peano
induction axiom is
For every set K, if 0 is in K and for every natural number x in K, x+1
is also in K, then every natural number is in K.
Is this the axiom that you are using?
--
David Marcus
herbzet
Phil wrote:
> Let's see if I can make this clear for you. I am 49 years old, almost
> 50, I have an incurable type of cancer called follicular non-Hodgkin's
> lymphoma that has an average life expectancy of 7 years, and I've had it
> 7 years. Older males -- I assume you are a young man -- have a rule of
> thumb, which they never say to anyone. As soon as they figure out that
> person X is going to play debate games, instead of having honest
> discussions, or repeat things told in private, or similar things, they
> cut off person X, without ever telling person X. I don't have time to
> play games, literally. I live on hydrocodone as it is, have chemo-brain,
> and I simply don't have time for it.
Please allow me to express my regret for your medical condition.
I hope you will be able to continue without too much discomfort
for many years to come.
I would like to assure you that the responses you have gotten
here in sci.logic have been made in good faith. People here
like arguments that can be, in principle at least, put into a
formal language, and demonstrated from premises, likewise formalized.
This cannot always be done, of course, even in principle, but
it's sort of the ideal that is pursued. Your idea that some
problems in math come from covert assumptions about actual
and potential infinity that do not match up with each other
is not, a priori, without merit, but what is really wanted
here is to put your ideas in a rigorous symbolic form, to
which the machinery of logic can then be applied.
Not so easy, of course, but very persuasive if it can be
done. Best of luck to you.
--
hz
Phil
Thanks, and I agree that many are "straight up" in their complete
disagreement with me! It's the ones who are either rabid defenders of
the great beliefs, or who just want to play debate games that annoy me.
It would indeed be nice to get rigorous terms, symbols, development,
etc., of potential versus actual infinity -- but then, perhaps the lack
of those things is why certain paradoxes currently exist?
Phil
Phil
> Phil wrote:
>
>>David Marcus wrote:
>
> << snip >>
>
>>All
>>subsets of 1 through n that do not use all the elements from 1 through n
>>have fewer elements than the set from 1 through n. If you even THINK
>>about asking me "How do you know this," I swear I'll spit on you next
>>time I see you. Well, okay, you can get the set 1 thru n by combining
>>two sets, the unused elements plus the used elements. I think even you
>>can take it from there. "How do you know you're not dreaming?" Enough.
>>
>>Since every subset of 1 thru n contains =< elements, it follows that we
>>only need to worry about the sets 1 thru n when it comes to maximum
>>size.
>
>
> What about the unbounded sets? Why don't we have to worry about them?
Define "unbounded sets," and specifically, contrast the elements that
the "unbounded sets" in N have access to that the sets "1 thru n" do not
have access to.
>
>
>>Since the set 1 thru n is finite for EVERY n -- and since no
>>subset of 1 thru n contains more elements than 1 thru n -- there is no
>>subset of N (the set of all natural numbers) that contains infinitely
>>many numbers.
>
>
> I suppose you mean no bounded set contains infinitely many numbers. But,
> you haven't said anything about the unbounded sets.
What you refer to as a "bounded set" has access to element of N as far
as I know. Perhaps you disagree?
>
>
>>>What principles are you taken as given regarding the natural numbers?
>>>The Peano axioms?
>>
>>Yes, although finite induction obviously is, not the "weak" point, but
>>the "fuzzy" point. And of course, its "fuzziness" exists in the area of
>>infinity. Technically, I see finite induction as perfectly valid given
>>the assumption of potential infinity. For example, in the Achilles, for
>>any point x a finite distance from the door, x/2 will also be a finite
>>distance from the door. Under finite induction, therefore, we can never
>>leave the room, and this is correct as long as we are limited to
>>potential infinity in terms of both the rate at which we cross halfway
>>points (points per second), and the amount of time that we are allowed
>>to cross those points (less than an infinite amount of time).
>
>
> I don't know what you mean by "finite induction". The usual Peano
> induction axiom is
>
> For every set K, if 0 is in K and for every natural number x in K, x+1
> is also in K, then every natural number is in K.
>
> Is this the axiom that you are using?
>
It is indeed. Should we add that it does not apply to the "unbounded
sets" of the natural numbers?
Chris Menzel
said:
> ...
> Thanks, and I agree that many are "straight up" in their complete
> disagreement with me! It's the ones who are either rabid defenders of
> the great beliefs, or who just want to play debate games that annoy
> me. It would indeed be nice to get rigorous terms, symbols,
> development, etc., of potential versus actual infinity -- but then,
> perhaps the lack of those things is why certain paradoxes currently
> exist?
I applaud your efforts to question and probe basic assumptions in
arithmetic and set theory, but I believe it's the other way round --
what you perceive to be paradoxes rely entirely upon the vagueness and
ambiguity inherent in any attempt at reasoning with informal, undefined
mathematical concepts, especially the notoriously slippery concept of
the infinite. If you were to cast those concepts into a rigorous form,
I think you would find that the paradoxes you see would vanish. I mean
no disrespect, but it honestly seems to me that your arguments resonate
with the very problems that hindered the development of a genuine
mathematics of the infinite prior to Cantor's illuminating definitions
of ordinality, equinumerosity, and cardinality, much as developments in
analysis were hindered prior to its arithmetization by the likes of
Cauchy, Bolzano, and Weierstrass. In particular, your arguments
involving finitude and precedence in an ordering seem to me to be
traceable to the absence of a theory of order types -- now a basic tool
of the study of sets and orderings. I believe you would receive a very
fair hearing in this forum if you were to attempt to make your reasoning
more explicit by starting with actual definitions of your basic concepts
and arguing step by step from those starting points. You are simply not
doing mathematics until you do that, and it is impossible to make clear
sense of your assertions.
David Marcus
> David Marcus wrote:
> > Phil wrote:
> >>David Marcus wrote:
> >
> > << snip >>
> >
> >>All
> >>subsets of 1 through n that do not use all the elements from 1 through n
> >>have fewer elements than the set from 1 through n. If you even THINK
> >>about asking me "How do you know this," I swear I'll spit on you next
> >>time I see you. Well, okay, you can get the set 1 thru n by combining
> >>two sets, the unused elements plus the used elements. I think even you
> >>can take it from there. "How do you know you're not dreaming?" Enough.
> >>
> >>Since every subset of 1 thru n contains =< elements, it follows that we
> >>only need to worry about the sets 1 thru n when it comes to maximum
> >>size.
> >
> > What about the unbounded sets? Why don't we have to worry about them?
>
> Define "unbounded sets,"
A subset S of N is /bounded/ if there is an m in N such that for all n
if n in S, then n < m.
If S is not bounded, then it is /unbounded/.
An example of an unbounded set is N itself. Another example is the even
natural numbers.
> and specifically, contrast the elements that
> the "unbounded sets" in N have access to that the sets "1 thru n" do not
> have access to.
Sorry. I don't know what you mean by "have access to". Can you define
this phrase?
> >>Since the set 1 thru n is finite for EVERY n -- and since no
> >>subset of 1 thru n contains more elements than 1 thru n -- there is no
> >>subset of N (the set of all natural numbers) that contains infinitely
> >>many numbers.
> >
> > I suppose you mean no bounded set contains infinitely many numbers. But,
> > you haven't said anything about the unbounded sets.
>
> What you refer to as a "bounded set" has access to element of N as far
> as I know. Perhaps you disagree?
Sorry. I don't know what you mean by "has access to".
> > I don't know what you mean by "finite induction". The usual Peano
> > induction axiom is
> >
> > For every set K, if 0 is in K and for every natural number x in K, x+1
> > is also in K, then every natural number is in K.
> >
> > Is this the axiom that you are using?
>
> It is indeed. Should we add that it does not apply to the "unbounded
> sets" of the natural numbers?
The usual axiom applies to all sets K of natural numbers. Are you using
the usual axiom?
--
David Marcus
Phil
wrong, but who cannot give good reasons why. As GD once said (several
years ago, and this is from memory, not a quote), Phil's probably wrong,
but the "responses" to his paradoxes seem to fall under two categories;
(1) they assert that "you just can't do that" [e.g., look at one of the
numbers on the natural number line that is preceded by infinitely many
numbers], or (2) they "correct" issues that Phil isn't even talking
about. However, you obviously honestly don't see it that way, which I
can hardly complain about.
I'll ask a couple of quick questions, just to make sure I'm not leaving
out something obvious. If there is a number, say (alepho - 1), that is
preceded by infinitely many numbers on the natural number line, but is
finite, why isn't the fact that the number's ordinal, which is finite,
doesn't match the number of numbers preceding it, which is infinite, a
problem?
Also, how can 10 multiplied by itself a finite number of times produce
an infinite number (10 to the number of digits in the natural numbers --
finite, since all numbers are finite -- equals the number of numbers)?
I'm guessing you view these as errors of interpretation, problems which
don't really follow from the basic premises and characteristics of
natural numbers, but I thought I would ask ...
Phil
David Marcus
> Chris Menzel wrote:
You've been telling people about these "paradoxes" for several years?
> I'll ask a couple of quick questions, just to make sure I'm not leaving
> out something obvious. If there is a number, say (alepho - 1), that is
> preceded by infinitely many numbers on the natural number line, but is
> finite, why isn't the fact that the number's ordinal, which is finite,
> doesn't match the number of numbers preceding it, which is infinite, a
> problem?
Because there isn't a natural number that is preceded by infinitely many
numbers.
> Also, how can 10 multiplied by itself a finite number of times produce
> an infinite number (10 to the number of digits in the natural numbers --
> finite, since all numbers are finite -- equals the number of numbers)?
It doesn't. 10 multiplied by itself a finite number of times is a finite
natural number. ("Finite natural number" is redundant since all natural
numbers are finite by definition.)
--
David Marcus
Phil
Well, I can try to clarify it ;-) What elements within the set of
natural numbers could an unbounded set (within the natural numbers, of
course) have, or contain (which is what I meant by "have access to"),
that sets containing numbers from "1 thru n," where n can be any of the
natural numbers, cannot contain. In other words, are there any natural
numbers in these unbounded sets that do NOT exist in any of the sets
from 1 thru n?
>
>
>>>>Since the set 1 thru n is finite for EVERY n -- and since no
>>>>subset of 1 thru n contains more elements than 1 thru n -- there is no
>>>>subset of N (the set of all natural numbers) that contains infinitely
>>>>many numbers.
>>>
>>>I suppose you mean no bounded set contains infinitely many numbers. But,
>>>you haven't said anything about the unbounded sets.
>>
>>What you refer to as a "bounded set" has access to element of N as far
>>as I know. Perhaps you disagree?
>
>
> Sorry. I don't know what you mean by "has access to".
>
>
>>>I don't know what you mean by "finite induction". The usual Peano
>>>induction axiom is
>>>
>>>For every set K, if 0 is in K and for every natural number x in K, x+1
>>>is also in K, then every natural number is in K.
>>>
>>>Is this the axiom that you are using?
>>
>>It is indeed. Should we add that it does not apply to the "unbounded
>>sets" of the natural numbers?
>
>
> The usual axiom applies to all sets K of natural numbers.
Would "all sets K" include all unbounded sets, where every element in
the unbounded set is a natural number, or do unbounded sets contain
elements that are not in "all sets K"?
Are you using
> the usual axiom?
I just said yes, but I'll say it again, yes.
>
G. Frege
>
> If there is a number [...] that is preceded by infinitely many
> numbers on the natural number line, but is finite, ...
>
There is no such number on the _nn line_.
>
> Also, how can 10 multiplied by itself a finite number of times
> produce an infinite number ...
>
It can't.
G. Frege
>
> In other words, are there any natural numbers in these unbounded sets
> that do NOT exist in any of the sets from 1 thru n?
>
Since for any n e IN the number n is in the "set from 1 thru n", for any n
e IN there is a "set from 1 thru n" such that n is in it. BUT that does not
mean that there is "set from 1 thru n" such that any n e IN is in it.
Proof: For each and any "set from 1 thru n" n+1 is not in the set. Hence
for any "set from 1 thru n" there is a number which is not in it. On the
other hand there is no (natural) number which is not in IN (by definition).
>>>>
>>>> I don't know what you mean by "finite induction". The usual Peano
>>>> induction axiom is
>>>>
>>>> For every set K, if 0 is in K and for every natural number x in K, x+1
>>>> is also in K, then every natural number is in K.
>>>>
>>>> Is this the axiom that you are using?
>>>
>>> It is indeed.
>>>
Fine.
>>
>> Should we add that it does not apply to the "unbounded sets" of the
>> natural numbers?
>>
No, we shouldn't. Since it _does_ apply to any set (K).
>
> Would "all sets K" include all unbounded sets [of natural numbers] ...
>
Yes.
David Marcus
> David Marcus wrote:
> > Phil wrote:
> >>David Marcus wrote:
> >>>Phil wrote:
> >>>>David Marcus wrote:
> >
> > Sorry. I don't know what you mean by "have access to". Can you define
> > this phrase?
>
> Well, I can try to clarify it ;-) What elements within the set of
> natural numbers could an unbounded set (within the natural numbers, of
> course) have, or contain (which is what I meant by "have access to"),
> that sets containing numbers from "1 thru n," where n can be any of the
> natural numbers, cannot contain. In other words, are there any natural
> numbers in these unbounded sets that do NOT exist in any of the sets
> from 1 thru n?
There are not. If n is a natural number, then n is in the set {1,..,n}
(for example). This doesn't mean that there is a natural number n such
that all natural numbers are in the set {1,..,n}.
(For this discussion, I'm assuming we are starting the natural numbers
with 1 rather than 0.)
> >>>I don't know what you mean by "finite induction". The usual Peano
> >>>induction axiom is
> >>>
> >>>For every set K, if 0 is in K and for every natural number x in K, x+1
> >>>is also in K, then every natural number is in K.
> >>>
> >>>Is this the axiom that you are using?
> >>
> >>It is indeed. Should we add that it does not apply to the "unbounded
> >>sets" of the natural numbers?
> >
> > The usual axiom applies to all sets K of natural numbers.
>
> Would "all sets K" include all unbounded sets, where every element in
> the unbounded set is a natural number, or do unbounded sets contain
> elements that are not in "all sets K"?
The first of your two choices, i.e., K can be an unbounded set of
natural numbers.
--
David Marcus
George Dance
> George Dance wrote:
> > On Mar 5, 5:50 pm, Phil <toob-head...@sbcglobal.net> wrote:
>
> >>G. Frege wrote:
>
> >>>On Mon, 05 Mar 2007 18:01:20 GMT, Phil <toob-head...@sbcglobal.net> wrote:
>
> >>>It doesn't. There are infinitely many natural numbers, but each and any is
> >>>finite.
>
> >>>I'll agree that this MIGHT _sound_ strange. Maybe the following formulation
> >>>is easier to digest:
>
> >>> Each and any natural number is finite.
>
> >>> But the set of _all_ natural numbers is not bound.
> >>> (I.e. it does not have an upper boundary.) With other
> >>> words, there is no biggest natural number. (I guess
> >>> that's immediately clear, no? If W would be this number,
> >>> W+1 would be another natural number, but bigger. Which
> >>> is an aburdity.)
>
> >>>F.
>
> >>George, this looks useful and workable; I'll get back to it later!
>
> > His name isn't George, BTW. He uses the name "G. Frege" (from Gottlob
> > Frege, the inventor of predicate logic) as an alias; for all anyone
> > knows, he may believe that he is Gottlob Frege.
>
> > .
>
> Oh, thanks! I was just thinking we have too damn many "Georges" around
> here to keep track of, and that will help!-
No problem. Now, just one more bit of help, and I'm outta here. (I
have my own trolls, on my own lists, to battle. 8)
You might have noticed that some of the people "discussing" with you
keep mentioning the name 'Mueckenheim'. That was a new one to me,
since I'd never heard of the guy; but it sounded important, so I
decided to do a search. Much to my surprise I found that he's not the
non-math crank I expected, but a published author. Here's a short
summary of his views, and the information on his book. (I realize
that may be of no use to you now, since you can't read German;
however, translations is a technical business.)
<quote>
Mueckenheim)
My goal is to show that there is no infinity (other than potential
infinity) at all. So we always remain in the finite domain. What
would
be the use of infinite natural numbers? What could you address by oo
=
1 + oo? My goal is to show that set theory is self contradictory, in
particular Cantor's claim of the infinite set of finite naturals.
This
goal is not difficult to achieve. There are several proofs. What is
much more difficult than I ever imagined is to have the set theorists
see and accept them.
By the way, a proof that the irrationals are not uncountable is
already
established by the observation that there is no pair of irrational
numbers which is not separated by a terminating rational number.
Regards, WM </q>
http://sci4um.com/about2-asc-2565.html
Book:
Die Mathematik des Unendlichen
Wolfgang Mückenheim
Shaker Verlag, Aachen, Germany, 2006, ISBN 3-8322-5587-7.
David Marcus
> You might have noticed that some of the people "discussing" with you
> keep mentioning the name 'Mueckenheim'. That was a new one to me,
> since I'd never heard of the guy; but it sounded important, so I
> decided to do a search.
You can find his posts in sci.math. He's been repeating the same
nonsense for a couple of years. He also has some wonderfully nonsensical
papers that he's posted to the arXiv.
> Much to my surprise I found that he's not the
> non-math crank I expected, but a published author.
He self-published his own book. He most definitely is a crank.
> <quote>
> Mueckenheim)
>
> My goal is to show that there is no infinity (other than potential
> infinity) at all. So we always remain in the finite domain. What
> would
> be the use of infinite natural numbers? What could you address by oo
> =3D
> 1 + oo? My goal is to show that set theory is self contradictory, in
> particular Cantor's claim of the infinite set of finite naturals.
> This
> goal is not difficult to achieve. There are several proofs. What is
> much more difficult than I ever imagined is to have the set theorists
> see and accept them.
>
> By the way, a proof that the irrationals are not uncountable is
> already
> established by the observation that there is no pair of irrational
> numbers which is not separated by a terminating rational number.
That's a good one. So, the fact that the rationals are dense implies the
reals are countable. I wonder how all the mathematicians missed that
theorem.
--
David Marcus
G. Frege
wrote:
>
> Much to my surprise I found that he's not the non-math crank I expected,
>
Sure he is.
"Wolfgang Mückenheim is a classic crank. Why do you imagine, as
you seem to do, that there is any point arguing with him? It
may seem odd that he has a job as a professor in technical and
mathematical subjects, but such things happen."
(Torkel Franzen)
G. Frege
<David...@alumdotmit.edu> wrote:
>>
>> Much to my surprise I found that he's not the
>> non-math crank I expected, but a published author.
>>
> He self-published his own book. He most definitely is a crank.
>
Sure he is a crank.
"My goal is to show that set theory is self contradictory,
in particular Cantor's claim of the infinite set of finite
naturals. This goal is not difficult to achieve. There are
several proofs." (WM)
"This goal is not difficult to achieve"? Sure if one is completely
incompetent concerning mathematics --- sounds EXTREMELY reasonable!
>>
>> By the way, a proof that the irrationals are not uncountable is
>> already established by the observation that there is no pair of
>> irrational numbers which is not separated by a terminating rational
>> number.
>>
> That's a good one. So, the fact that the rationals are dense implies the
> reals are countable. I wonder how all the mathematicians missed that
> theorem.
>
Well, they had to wait for Mückenheim (to come)!
herbzet
Well, it's intuititively plausible. If there's a rational between
every two irrationals, it would seem that the rationals and the
irrationals are equinumerous. Like in a picket fence, there
are as many spaces between pickets as there are pickets. Alright,
one less, unless the fence is a closed curve, or infinitely long.
This would take some careful explaning to sort out, and I'm not
the guy to do it, because I'm unfamiliar with that particular
proof, though I've seen it mentioned before.
--
hz
Virgil
wrote:
> "G. Frege" wrote:
> >
> > On Thu, 8 Mar 2007 14:37:29 -0500, David Marcus
> > <David...@alumdotmit.edu> wrote:
> >
> > >>
> > >> Much to my surprise I found that he's not the
> > >> non-math crank I expected, but a published author.
> > >>
> > > He self-published his own book. He most definitely is a crank.
> > >
> > Sure he is a crank.
> >
> > "My goal is to show that set theory is self contradictory,
> > in particular Cantor's claim of the infinite set of finite
> > naturals. This goal is not difficult to achieve. There are
> > several proofs." (WM)
> >
> > "This goal is not difficult to achieve"? Sure if one is completely
> > incompetent concerning mathematics --- sounds EXTREMELY reasonable!
> >
> > >>
> > >> By the way, a proof that the irrationals are not uncountable is
> > >> already established by the observation that there is no pair of
> > >> irrational numbers which is not separated by a terminating rational
> > >> number.
> > >>
> > > That's a good one. So, the fact that the rationals are dense implies the
> > > reals are countable. I wonder how all the mathematicians missed that
> > > theorem.
>
> Well, it's intuititively plausible. If there's a rational between
> every two irrationals, it would seem that the rationals and the
> irrationals are equinumerous.
Possibly at first glance, but that assumes that the number of rationals
between two irrationals is the same as the number of irrationals between
two rationals, which is distinctly not the case.
What IS the case, though it seems contradictory at first glance, is
(1) the number of irrationals between any two rationals is the same as
the number between any other two rationals, regardless of how far apart
they are, and
(2) the number of rationals between any two irrationals is the same as
the number between any other two irrationals, regardless of how far
apart they are.
Note, for example, that if a and b are rational, with a < b, then
x -> (b-a)x + a is an order preserving bijection of the reals to the
reals carrying rationals to rationals and irrationals to irrationals,
regardless of the distance between a and b.
George Dance
> On 8 Mar 2007 11:18:07 -0800, "George Dance" <georgedanc...@yahoo.ca>
> wrote:
>
>
>
> > Much to my surprise I found that he's not the non-math crank I expected,
>
> Sure he is.
>
> "Wolfgang Mückenheim is a classic crank. Why do you imagine, as
> you seem to do, that there is any point arguing with him? It
> may seem odd that he has a job as a professor in technical and
> mathematical subjects, but such things happen."
> (Torkel Franzen)
>
That has as much evidential value as:
"The Pot is black." (The Kettle)
G. Frege
wrote:
>>>
>>> Much to my surprise I found that he's not the non-math crank I expected,
>>>
>> Sure he is.
>>
>> "Wolfgang Mückenheim is a classic crank. Why do you imagine, as
>> you seem to do, that there is any point arguing with him? It
>> may seem odd that he has a job as a professor in technical and
>> mathematical subjects, but such things happen."
>>
>> (Torkel Franzen)
>>
> That has as much evidential value as [bla bla].
>
Huh? It's a simple fact that Mückenheim is a mathematical crank. Something
which would be _immediately_ clear to you if you weren't yourself rather
incompetent concerning mathematical topics (to say the least).
If you think otherwise you might present a SINGLE "argument" put forward by
WM ("proving that set theory is inconsistent") which is NOT completely non-
sensical (i.e. mathematical nonsense). Do it!
G. Frege
>>>>
>>>> By the way, a proof that the irrationals are not uncountable is
>>>> already established by the observation that there is no pair of
>>>> irrational numbers which is not separated by a terminating rational
>>>> number. [WM]
>>>>
>>> That's a good one. So, the fact that the rationals are dense implies the
>>> reals are countable. I wonder how all the mathematicians missed that
>>> theorem. [Virgil]
>>>
> Well, it's intuititively plausible.
>
Exactly, or rather it _may_ be plausible at first glance.
Line of thought:
>
> If there's a rational between every two irrationals, it would seem that
> the rationals and the irrationals are equinumerous [+/-1]. Like in a
> picket fence, there are as many spaces between pickets as there are pickets.
>
Right. (Note that you _chose_ two countable set's here! Hence they are
indeed equinumerous! :-)
And there's an obvious ("natural") bijection, of course:
any picket is related with the space immediately following.
(Note that concerning Q and R there is no such "natural
bijection", since for any numbers in R and Q there are no
elements "next to it".)
>
> This would take some careful explaining to sort out, and I'm not
> the guy to do it, because I'm unfamiliar with that particular
> proof, though I've seen it mentioned before.
>
Your "argument" from above doesn't work for R and Q. It's true: between any
two (different) numbers e Q there is a number e R. And between any two
(different) numbers e R there is a number e Q. BUT (in contrast to the
example you mentioned above) there are _infinitely many_ numbers e Q
between any two numbers e R and there are _infinitely many_ numbers e R
between two numbers e Q. Hence the nice "picture" you considered above
"breaks down."
( In contrast to your picket fence ...-picket-space-picket-...,
h e r e there are no immediate "successors" of the considered
objects. )
Actually, it's completely misleading since there are _uncountable many_
numbers e R between any two numbers e Q (but only countable many numbers e
Q between any two numbers e R).
With other words, the fact that between any two (different) numbers e Q
there is a number e R, and between any two (different) numbers e R there is
a number e Q, does (in contrast to the example you considered) does _not_
lead to a "natural bijection" between Q and R.
Again we see: Looking plausible (at first glance) is not good enough in
mathematics. (Hence we rely on proofs.)
G. Frege
>
> Your "argument" from above doesn't work for R and Q. It's true: between any
> two (different) numbers e Q there is a number e R. And between any two
> (different) numbers e R there is a number e Q. BUT (in contrast to the
> example you mentioned above) there are _infinitely many_ numbers e Q
> between any two numbers e R and there are _infinitely many_ numbers e R
> between two numbers e Q. Hence the nice "picture" you considered above
> "breaks down."
>
> ( In contrast to your picket fence ...-picket-space-picket-...,
> h e r e there are no immediate "successors" of the considered
> objects. )
>
> Actually, it's completely misleading since there are _uncountable many_
> numbers e R between any two numbers e Q (but only countable many numbers e
> Q between any two numbers e R).
>
Note though (especially the last sentence)...
"The irrational numbers may just be defined as the set of gaps in Q. Since
the set of the irrational numbers has the cardinal aleph, in our case the
set of the gaps has a greater cardinal than Q itself; this phenomenon is
not surprising since the cuts are essentially subsets and not members of
the set. By mere spatial intuition or classical geometrical attitudes
without set-theoretical methods, the phenomenon can neither be stated nor
comprehended." (Fraenkel, Levy)
George Dance
> On 11 Mar 2007 06:43:19 -0700, "George Dance" <georgedanc...@yahoo.ca>
> wrote:
>
>
>
> >>> Much to my surprise I found that he's not the non-math crank I expected,
>
> >> Sure he is.
>
> >> "Wolfgang Mückenheim is a classic crank. Why do you imagine, as
> >> you seem to do, that there is any point arguing with him? It
> >> may seem odd that he has a job as a professor in technical and
> >> mathematical subjects, but such things happen."
>
> >> (Torkel Franzen)
> > That has as much evidential value as
<unsnip>
"The Pot is black." (The Kettle)
</us>
> If you think otherwise you might present a SINGLE "argument" put forward by
> WM ("proving that set theory is inconsistent") which is NOT completely non-
> sensical (i.e. mathematical nonsense). Do it!
>
Sorry, Mr. Argument-from-ignorance. It's not my job to prove that
he's not a crank. If you make a claim, the onus is on you to back it
up; which you don't do by quoting Torkel Franzen's old flames.
.
Phil
> On Thu, 08 Mar 2007 00:34:02 GMT, Phil <toob-h...@sbcglobal.net> wrote:
>
>
>>In other words, are there any natural numbers in these unbounded sets
>>that do NOT exist in any of the sets from 1 thru n?
>>
>
> Since for any n e IN the number n is in the "set from 1 thru n", for any n
> e IN there is a "set from 1 thru n" such that n is in it. BUT that does not
> mean that there is "set from 1 thru n" such that any n e IN is in it.
>
> Proof: For each and any "set from 1 thru n" n+1 is not in the set. Hence
> for any "set from 1 thru n" there is a number which is not in it. On the
> other hand there is no (natural) number which is not in IN (by definition).
Ridiculous, look at your "reasoning," if I may be forgiven for so
abusing the word. You used an argumentative structure to show that for
any set 1 thru n, there is an element n+1 that is not in the set. Fine,
duh, but you MUST then use that same argumentative structure to see if
you can "prove" just the reverse, as in, for any element n, there is a
set 1 thru n+1 that contains an element larger than n. Equally fine,
equally duh, and I just "proved" that there are sets in IN which contain
elements larger than any n, which is no less stupid than the conclusion
from your proof, but which does have the advantage of being more
obviously stupid. And just to make sure that you do not attempt to
continue to be stupid with your comment that "On the other hand there is
no (natural) number" -- for example, n -- "which is not in IN (by
definition)," I can just as easily say that "for every (natural) number
-- as in "n" -- in the set of IN, there exists a subset which contains 1
thru that number -- as in "n" -- by the definition of sets."
That's the problem with modern mathematicians, and scientists in
general, they confuse argument with proof. Do you know WHY you must use
the same process of reasoning, the same argumentative structure (surely
there's a better term) in reverse, or in an alternate manner? A proof,
as opposed to a mere argument, is something that NECESSARILY FOLLOWS
from a set of premises. That means, among other things, that there are
no other conclusions which can also follow from the same set of
premises. If you can find two or more contradictory conclusions from a
set of premises, then you cannot trust any of them. As a side note, the
existence of several conclusions proves that either (1) the
argumentative structure is flawed, which is probably the case with your
argumentative structure (2) the premises are mutually contradictory, in
the sense that they are too restrictive (3) the premises are not
restrictive enough, meaning that they allow several possibilities, and
you need another premise, the parallel postulates being the classic example.
Now, SOMETIMES you can work hard, and find additional TRUE proofs that
will allow you to narrow down the results to just one conclusion, but
one way or another, until you can get a conclusion which NECESSARILY
FOLLOWS from your premises, you have nothing.
Phil
Phil
Well! That does indeed sound extremely familiar! I guess I need to check
out his proofs and see if any are the same as mine (3 or so; I could
swear there was another, but it's been so long ...). I think, in
contrast, that there must be actual as well as potential infinity, but
that does not mean that his criticisms of "infinitely many finite
numbers" is flawed, or for that matter the idea of an internally
consistent realm limited by potential infinity.
Yes, my own conclusion is that whatever the problems are, they are so
wound up in every aspect of mathematics that set theorists use the first
5 areas to defend the 6th, then the last 5 to defend the first, etc.,
meaning it's going to take a LOT OF WORK to undo this mess. Thanks, I'll
at least try to check this out.
Phil