Relativity of Arithmetic of the Natural Numbers.
Nam Nguyen
(*) There exist 2 prime numbers p1 p2 in which the
statement "p1 is less than p2" is relativistic
(i.e. subjective)
is true ?
[I know that sounds "crazy", but let's think of an analogy
"c(t1) < c(t2)" where c is a Choice function.]
Marshall
> Would you believe that this meta statement:
>
> (*) There exist 2 prime numbers p1 p2 in which the
> statement "p1 is less than p2" is relativistic
> (i.e. subjective)
>
> is true ?
Can you be more explicit about what you mean by
relativistic/subjective? I'm assuming of course that
you are using the usual meanings of "prime"
and "less than."
Marshall
Nam Nguyen
Sure. First, yes, I do mean the usual semantic of "prime" and
"less than". On the the question of the meaning of "relative/
subjective", in this context, if we're given the _existence_
of a fixed universe-set U and there's no way at all we can know
how to differentiate 2 different models then some model truths
would be relative, in the sense it's entirely subjective to, say,
you and me to claim which model is the one in which a certain
formula is true, or false and the other model the other way around!
That's an overview of the situation. Let me present some more
details here.
***
Let rN (Relativistic Natural Number) be a 1st order system written
in a particular language L = L(rN) = L(0, S, <, +, *, <', +', *',
<", +", *",...). In other words, any of the following sub-languages
of L:
(0, S, <, +, *,), (0, S', <', +', *'), (0, S", <", +", *"), ...
would form what we'd refer to as "the standard language of arithmetic".
NOTE: the symbol '0' is common in all those sub languages.
The axioms of rN would be described as below:
Given any sub language above, we would have Q's axioms in that sub
language as axioms of rN.
Now, the numerals of rN are:
- 0
- S0, SS0, SSS0, ...
- S'0, S'S'0, S'S'S'0, ...
- S"0, S"S"0, S"S"S"0, ...
...
In FOL=, for sure 0=0 but what about S0=S'0, or S0=S"0, or S'0=S"0,
...? The answer is if you believe there _exist_ a universe U from which
a model M of rN would arise (i.e. would be interpreted) then it's
entire up to any of us (you and I e.g.) - hence the sense of being
relative or subjective - to say whether or not any 2 syntactically
isomorphic numerals (other that 0 itself) would be _existentially_
the same.
For instance, we both at the same time _subjectively_ let SSS0=S'S'S'0
while SSSSS0 and S"S"S"S"S"0 symbolize 2 different existences, in all
that however SSS0, S'S'S'0, SSSSS0, S"S"S"S"S"0 still symbolize the
common property of being a prime number!
It's in the sense that for any 2 given sub languages of L(rN), in general,
any 2 syntactically isomorphic numerals could be _subjectively_ interpreted
as existentially identical or different that the merit of the meta statement
(*) comes from.
[One could certainly use a ZFC model to demonstrate this more clearly
and I might do this in a later post.]
Marshall
Like this?
Suppose M is the usual model of the natural numbers. Suppose
further I have two additional different models of the natural numbers,
M' and M'', and no way to inspect them relative to each other, and
model M' happens to use the usual sequence:
0, s'0=1, s's'0=2, s's's'0=3, s's's's'0=4 ...
while model M'' uses the sequence except with 2 and 3 swapping
positions:
0, s''0=1, s''s''0=3, s''s''s''0=2, s''s''s''s''0=4 ...
then in M', we have
s's'0 <' s's's'0 (which is to say, 2 <' 3)
and in M'', we have
s''s''0 <'' s''s''s''0 (which is to say, 3 <'' 2)
So if you give me a model and I have no way to inspect
it, I can't tell if whatever element of the model maps to 2
is less than or greater than whatever element of the model
maps to 3.
Marshall
RussellE
>
> - Show quoted text -
There are some exceptions.
Consider ss0 and s's'0 and let ss0=2 in the "standard" model.
I can say ss0 <= s's'0 for all models where s's'0 is prime.
I can also say (sss0 <= s's's'0) OR (s's's'0 = ss0) (assuming
s's's'0 is prime).
Russell
- 2 many 2 count
Nam Nguyen
Very close but not quite 100% of what I had said. I'm going to explain that.
>
> Suppose M is the usual model of the natural numbers. Suppose
> further I have two additional different models of the natural numbers,
> M' and M'', and no way to inspect them relative to each other, and
> model M' happens to use the usual sequence:
>
> 0, s'0=1, s's'0=2, s's's'0=3, s's's's'0=4 ...
>
> while model M'' uses the sequence except with 2 and 3 swapping
> positions:
>
> 0, s''0=1, s''s''0=3, s''s''s''0=2, s''s''s''s''0=4 ...
>
> then in M', we have
>
> s's'0 <' s's's'0 (which is to say, 2 <' 3)
>
> and in M'', we have
>
> s''s''0 <'' s''s''s''0 (which is to say, 3 <'' 2)
>
>
> So if you give me a model and I have no way to inspect
> it, I can't tell if whatever element of the model maps to 2
> is less than or greater than whatever element of the model
> maps to 3.
First, we should be careful and not get "entangled" with the notations.
Let's leave '0' be the symbol of L(rN) as before and without loss of
generality let's let the _set_ of the naturals be defined as a ZF set
as:
N = {e0 = {}, e1 = {e0}, e2 = {e1}, e3 = {e2}, ...}
Secondly, it's crucial to note that in and of itself *N is just a set*.
To call N as the naturals, more ZF sets have to be taken in account and
one of which would be the set of ordered-pairs symbolized by any of the
unary function symbol S, S', S", .... Without loss of generality, such
an ordered-pair set could be presented as any of the following sequences:
(1) e0 -> e1 -> e2 -> e3 -> e4 -> e5 -> e6 -> e7 -> e8 -> ....
(2) e0 -> e1 -> e3 -> e2 -> e4 -> e5 -> e6 -> e7 -> e8 -> ....
(3) e0 -> e1 -> e2 -> e5 -> e4 -> e7 -> e6 -> e3 -> e8 -> ....
...
Let's note that in the 1st 3 sequences, e2, e3, e5, e7 have the identical
semantic of being a prime. Yet the semantic "less than" of any pair of those
4 elements is subjective/relative: to which unary symbol any of us has
subjectively taken to be _the chosen one_.
My original claim is based on such observation.
Marshall
>
> Let's note that in the 1st 3 sequences, e2, e3, e5, e7 have the identical
> semantic of being a prime. Yet the semantic "less than" of any pair of those
> 4 elements is subjective/relative: to which unary symbol any of us has
> subjectively taken to be _the chosen one_.
It's also the case that whether they're prime or not depends on
which sequence is chosen.
Really, I don't see that you're doing anything other than
renaming the naturals. I don't see how anything interesting
is likely to result.
Suppose we have the usual 2-valued boolean algebra,
and also an algebra of two elements in which the top
and bottom elements are called "male" and "female."
In that case, whether "x ^ ~x" is "female" or not depends
on which algebra we are interpreting it in.
Marshall
Nam Nguyen
> On Dec 11, 9:59 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Let's note that in the 1st 3 sequences, e2, e3, e5, e7 have the identical
>> semantic of being a prime. Yet the semantic "less than" of any pair of those
>> 4 elements is subjective/relative: to which unary symbol any of us has
>> subjectively taken to be _the chosen one_.
>
> It's also the case that whether they're prime or not depends on
> which sequence is chosen.
Correct.
>
> Really, I don't see that you're doing anything other than
> renaming the naturals. I don't see how anything interesting
> is likely to result.
Note that in an rN model, all of these natural-number sequences
begin with a common element - e0. And that's interesting because
you can't tell which one is "the most _natural_" sequence without
your subjective choice!
In other words, anyone's speaking of "the naturals" is quite a subjective/
relative affair.
>
> Suppose we have the usual 2-valued boolean algebra,
> and also an algebra of two elements in which the top
> and bottom elements are called "male" and "female."
> In that case, whether "x ^ ~x" is "female" or not depends
> on which algebra we are interpreting it in.
That's correct. And while in such a finite case it's so trivial
to the point of being unimpressive a notion, in the case when
an omega property is involved, it would be quite a notion!
GC could be such omega property. In any rate, it wouldn't be too
hard to demonstrate that there's always one such omega property
about the naturals.
Marshall
> Marshall wrote:
> > On Dec 11, 9:59 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> Let's note that in the 1st 3 sequences, e2, e3, e5, e7 have the identical
> >> semantic of being a prime. Yet the semantic "less than" of any pair of those
> >> 4 elements is subjective/relative: to which unary symbol any of us has
> >> subjectively taken to be _the chosen one_.
>
> > It's also the case that whether they're prime or not depends on
> > which sequence is chosen.
>
> Correct.
>
> > Really, I don't see that you're doing anything other than
> > renaming the naturals. I don't see how anything interesting
> > is likely to result.
>
> Note that in an rN model, all of these natural-number sequences
> begin with a common element - e0. And that's interesting because
> you can't tell which one is "the most _natural_" sequence without
> your subjective choice!
>
> In other words, anyone's speaking of "the naturals" is quite a subjective/
> relative affair.
Anyone speaking of the "the" set of glyphs that name the naturals
is speaking about something subjective. Anyone speaking about
the naturals is speaking about a single uniquely defined thing.
A rose by any other name would still be the smallest inductive
set.
Renaming is not profound.
> > Suppose we have the usual 2-valued boolean algebra,
> > and also an algebra of two elements in which the top
> > and bottom elements are called "male" and "female."
> > In that case, whether "x ^ ~x" is "female" or not depends
> > on which algebra we are interpreting it in.
>
> That's correct. And while in such a finite case it's so trivial
> to the point of being unimpressive a notion, in the case when
> an omega property is involved, it would be quite a notion!
>
> GC could be such omega property. In any rate, it wouldn't be too
> hard to demonstrate that there's always one such omega property
> about the naturals.
What you call things is unimportant. Names don't work
the way you seem to think they do.
Marshall
Nam Nguyen
> On Dec 11, 10:36 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Marshall wrote:
>>> On Dec 11, 9:59 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>> Let's note that in the 1st 3 sequences, e2, e3, e5, e7 have the identical
>>>> semantic of being a prime. Yet the semantic "less than" of any pair of those
>>>> 4 elements is subjective/relative: to which unary symbol any of us has
>>>> subjectively taken to be _the chosen one_.
>>> It's also the case that whether they're prime or not depends on
>>> which sequence is chosen.
>> Correct.
>>
>>> Really, I don't see that you're doing anything other than
>>> renaming the naturals. I don't see how anything interesting
>>> is likely to result.
>> Note that in an rN model, all of these natural-number sequences
>> begin with a common element - e0. And that's interesting because
>> you can't tell which one is "the most _natural_" sequence without
>> your subjective choice!
>>
>> In other words, anyone's speaking of "the naturals" is quite a subjective/
>> relative affair.
>
> Anyone speaking of the "the" set of glyphs that name the naturals
> is speaking about something subjective.
> Anyone speaking about
> the naturals is speaking about a single uniquely defined thing.
That's where you've missed the point, Marshall. A model of a formal
system is a decision procedure in which each and every formula in the
language *should be asserted* as true, or false. If there's even a
possibility that it's impossible to assert the truth or falsehood
of a formula then the model can't be said to be unique, as a body
of truths, however one "thinks" one has precisely defined it!
You should realize that one could give a precise definition of a concept
and yet a statement about the concept could be still relativistic.
For instance, the definition of a polygon is precise and yet the
the statement "There exists a polygon with n vertexes, where n
is the smallest counter example of GC" could be relativistic to
*which definition of the naturals*, say, Marshall has subjectively
chosen to "uniquely" talk about!
To paraphrase a well known saying, in so far as one can uniquely
define the natural numbers, that definition is not about the
natural numbers!
> A rose by any other name would still be the smallest inductive
> set.
But listen to what Shoenfield said: "each expression shall be used
as a name for itself". So it's not the name "rose" which is an issue,
its the formula (which is an expression), e.g.,GC that is the issue
here.
>
> Renaming is not profound.
Apparently you're wrong.
>>> Suppose we have the usual 2-valued boolean algebra,
>>> and also an algebra of two elements in which the top
>>> and bottom elements are called "male" and "female."
>>> In that case, whether "x ^ ~x" is "female" or not depends
>>> on which algebra we are interpreting it in.
>> That's correct. And while in such a finite case it's so trivial
>> to the point of being unimpressive a notion, in the case when
>> an omega property is involved, it would be quite a notion!
>>
>> GC could be such omega property. In any rate, it wouldn't be too
>> hard to demonstrate that there's always one such omega property
>> about the naturals.
>
> What you call things is unimportant. Names don't work
> the way you seem to think they do.
You've missed the point entirely. The issue here in my op is
about arithmetic _truth_ being relative. It's not about the
relativity of "naming"!
Can you re-read the post and offer some analysis about the
mentioned relativity of arithmetic truth? I mean you and I are
capable of naming the naturals "rose", "tulips", "Bush", etc...
but what's the point in that?
Marshall
Sorry, but you are mistaken. If the objects in the
theory are renamed, the formulas must be renamed as
well. Normally we say "2<3" but if we exchange the
names of 2 and 3, then the names in the formula
changes as well. And yet it is still the same formula.
The concrete syntax may change but the abstract
syntax does not. The formula is interesting only
insofar as it is a representation of the abstract syntax.
The concrete syntax, and manipulations thereof,
are uninteresting.
What you are describing is relativity of concrete
syntax, not relativity of truth.
Marshall
Nam Nguyen
Really? Given that you've used dubious-to-incorrect buzzwords below
such as "objects in the theory", "abstract syntax", "relativity of
concrete syntax", in technically arguments, perhaps you should consider
refraining from calling anybody else but you as being mistaken.
[Note: none of the 5 textbooks I have would use those phrases the
way you do or would use them at all and, iirc, I've heard anyone
expressing as such for the past 10 years in the ng).
> If the objects in the
> theory are renamed, the formulas must be renamed as
> well. Normally we say "2<3" but if we exchange the
> names of 2 and 3, then the names in the formula
> changes as well.
You seemed to be clueless in what you said here. Technically speaking
(as opposed to your your expressions here), "2<3" is written in _a_
language of arithmetic such as "SS0 < SSS0" as _one_ formula. There's no
"exchange [of] the names of 2 and 3" whatsoever! [This is actually basic
knowledge of FOL].
> And yet it is still the same formula.
> The concrete syntax may change
Huh? How is it that "it is still the same [syntactical] formula" and
yet "The concrete syntax may change"? You're entirely mistaken on the
basic notion of what a formula be.
> but the abstract syntax does not.
Nobody but you would use the phrase "abstract syntax" in technically
refuting FOL-reasoning technical issues.
> The formula is interesting only
> insofar as it is a representation of the abstract syntax.
I suppose if we replace "the abstract syntax" by "the abstract concept"
then fine: that's what we use the concrete mathematical formulas for,
which is a trivial observation.
> The concrete syntax, and manipulations thereof,
> are uninteresting.
You're saying then the rules of inference in FOL aren't interesting!
No wonder why it's hard to follow your arguments so far.
> What you are describing is relativity of concrete
> syntax, not relativity of truth.
For what it's worth, as far as FOL is concerned, "concrete syntax" is
*not* relative - but truth is! You yourself can't even define what it
means by the "relativity of concrete syntax" of a formula to start
with! On the other hand, the truth of "1 + 1 = 0" is relative to what
model of the language you have subjectively chosen as the underlying
model.
Marshall
> Marshall wrote:
> > On Dec 12, 8:58 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> >> For instance, the definition of a polygon is precise and yet the
> >> the statement "There exists a polygon with n vertexes, where n
> >> is the smallest counter example of GC" could be relativistic to
> >> *which definition of the naturals*, say, Marshall has subjectively
> >> chosen to "uniquely" talk about!
>
> >> To paraphrase a well known saying, in so far as one can uniquely
> >> define the natural numbers, that definition is not about the
> >> natural numbers!
>
> >>> A rose by any other name would still be the smallest inductive
> >>> set.
> >> But listen to what Shoenfield said: "each expression shall be used
> >> as a name for itself". So it's not the name "rose" which is an issue,
> >> its the formula (which is an expression), e.g.,GC that is the issue
> >> here.
>
> >>> Renaming is not profound.
> >> Apparently you're wrong.
>
> >> about arithmetic _truth_ being relative. It's not about the
> >> relativity of "naming"!
>
> >> Can you re-read the post and offer some analysis about the
> >> mentioned relativity of arithmetic truth? I mean you and I are
> >> capable of naming the naturals "rose", "tulips", "Bush", etc...
> >> but what's the point in that?
>
> > Sorry, but you are mistaken.
>
> Really?
Yes.
> Given that you've used dubious-to-incorrect buzzwords below
> such as "objects in the theory", "abstract syntax", "relativity of
> concrete syntax", in technically arguments, perhaps you should consider
> refraining from calling anybody else but you as being mistaken.
> [Note: none of the 5 textbooks I have would use those phrases the
> way you do or would use them at all and, iirc, I've heard anyone
> expressing as such for the past 10 years in the ng).
I deeply apologize for your lack of familiarity with the
terms "abstract syntax" and "concrete syntax." Here,
let me help you:
http://justfuckinggoogleit.com/search.pl?query=abstract%20syntax
(Note that the introduction of the word "relativity" is yours.)
> > If the objects in the
> > theory are renamed, the formulas must be renamed as
> > well. Normally we say "2<3" but if we exchange the
> > names of 2 and 3, then the names in the formula
> > changes as well.
>
> You seemed to be clueless in what you said here. Technically speaking
> (as opposed to your your expressions here), "2<3" is written in _a_
> language of arithmetic such as "SS0 < SSS0" as _one_ formula. There's no
> "exchange [of] the names of 2 and 3" whatsoever! [This is actually basic
> knowledge of FOL].
If for a model you exchange the referents of "ss0" and "sss0" then
which of those referents is less than the other according to that
model changes. "SS0<SSS0" remains; what it is referring to has
changed. This is the only "relativity" you've uncovered, and
it's uninteresting.
> > And yet it is still the same formula.
> > The concrete syntax may change
>
> Huh? How is it that "it is still the same [syntactical] formula" and
> yet "The concrete syntax may change"? You're entirely mistaken on the
> basic notion of what a formula be.
>
> > but the abstract syntax does not.
> Nobody but you would use the phrase "abstract syntax" in technically
> refuting FOL-reasoning technical issues.
Alas, I must also deeply apologize for your lack of familiarity
with the concepts of "abstract syntax" and "concrete syntax"
as well as the terms themselves. Also note that I am saying
nothing about FOL-reasoning issues; neither are you.
> > The formula is interesting only
> > insofar as it is a representation of the abstract syntax.
>
> I suppose if we replace "the abstract syntax" by "the abstract concept"
> then fine: that's what we use the concrete mathematical formulas for,
> which is a trivial observation.
>
> > The concrete syntax, and manipulations thereof,
> > are uninteresting.
>
> You're saying then the rules of inference in FOL aren't interesting!
> No wonder why it's hard to follow your arguments so far.
Let me be more explicit: the syntactic embodiment of FOL
and FOL reasoning lies in the abstract syntax and the
semantics, and the relationship they have that was proven
in the completeness theorem.
Infinitely many choices exist for a concrete syntax for FOL's
abstract syntax; which choice we pick is uninteresting.
Of course the specific structure of that concrete syntax
has to match the abstract syntax, just as the abstract
syntax has to match the semantics; this does not change
the fact that the choice of the concrete syntax is
arbitrary and of no interest.
> > What you are describing is relativity of concrete
> > syntax, not relativity of truth.
>
> For what it's worth, as far as FOL is concerned, "concrete syntax" is
> *not* relative - but truth is!
Only the concrete syntax for truth is relative. The truth
itself, the semantics, is untouched by the choice of
concrete syntax. Likewise, the abstract syntax is also
untouched, unaffected, by the choice of concrete syntax.
> You yourself can't even define what it
> means by the "relativity of concrete syntax" of a formula to start
> with! On the other hand, the truth of "1 + 1 = 0" is relative to what
> model of the language you have subjectively chosen as the underlying
> model.
It is true that "1+1=0" has to be interpreted relative to some
concrete syntax. What is not true, however, is the suggestion
that anything interesting results. For whatever concrete
syntax we use, and for whatever formula we are discussing,
what we are *really* discussing is the referent of the string
of symbols for the formula; if you keep the string of symbols
the same and change the concrete syntax you are using, then
you are talking about a different formula.
That two different formulas might have different truth values
is of course true, and uninteresting. That one formula might
have two different sets of symbols that can express it is
of course true, and uninteresting. That a specific string of
symbols might refer to one formula in one concrete syntax
and a different formula in a different concrete syntax,
and that those formulas might have different truth values
is of course true, and uninteresting.
Marshall
Nam Nguyen
First, for your credibility sake, this is sci.log and the underlying
framework is FOL formal system and you couldn't just simply throw
any *other* technical terms "abstract syntax" from other fields and
waive a dubious URL and hope that what you babbled with those terms
would validate your ignorance of the FOL formal system subjects, without
your explaining some good details why those new terms are relevant
to in the context of FOL reasoning. [An intoxicated student could
amuse a conversation by throwing in a bunch of jargons unrelated
to the context at hand; but at least he's drunk!]
Secondly, be disciplined in making arguments. There, I was critical of
*your* "relativity of concrete syntax". What you yourself stated and
that you've *not explained* what the phrase means are *FACTS*. Respond
to the question about those facts, if you have any desire for your
argument to be credible. Whether or not somebody else's unfamiliarity
with "concrete syntax" isn't a fact you could use in your credibility.
Finally, you and I are free to introduce any terms ("relativity", "Bush",
..., e.g.) as we wish, as long as we _clearly_ define and use them in the
the context of FOL reasoning.
>>> If the objects in the
>>> theory are renamed, the formulas must be renamed as
>>> well. Normally we say "2<3" but if we exchange the
>>> names of 2 and 3, then the names in the formula
>>> changes as well.
>> You seemed to be clueless in what you said here. Technically speaking
>> (as opposed to your your expressions here), "2<3" is written in _a_
>> language of arithmetic such as "SS0 < SSS0" as _one_ formula. There's no
>> "exchange [of] the names of 2 and 3" whatsoever! [This is actually basic
>> knowledge of FOL].
>
> If for a model you exchange the referents of "ss0" and "sss0" then
> which of those referents is less than the other according to that
> model changes. "SS0<SSS0" remains; what it is referring to has
> changed. This is the only "relativity" you've uncovered, and
> it's uninteresting.
Each of us is free to believe what's interesting or not. The title
of the thread is "Relativity of Arithmetic of the Natural Numbers"
and my original meta statement/question about it is:
>>Would you believe that this meta statement:
>>
>>(*) There exist 2 prime numbers p1 p2 in which the
>> statement "p1 is less than p2" is relativistic
>> (i.e. subjective)
>>
>>is true ?
Are you saying here that (*) is true?
> It is true that "1+1=0" has to be interpreted relative to some
> concrete syntax. What is not true, however, is the suggestion
> that anything interesting results.
OK. Then, GC isn't that much a longer formula that "1+1=0" and
so would you find its truth's potential being relative, to which
"naturals" you've chosen, to be interesting or not?
> For whatever concrete
> syntax we use, and for whatever formula we are discussing,
> what we are *really* discussing is the referent of the string
> of symbols for the formula; if you keep the string of symbols
> the same and change the concrete syntax you are using, then
> you are talking about a different formula.
Again you've mixed up basic FOL concepts. If you "keep the string
of symbols the same" then you're talking about *the very same formula*.
Period. (Good grief!)
>
> That two different formulas might have different truth values
> is of course true, and uninteresting.
Where did anybody here state that *two different formulas* having
2 different truth values would be interesting? For instance, who
has suggested something like it's interesting that "0+0=0" is
true while "1+1=0" is false?
> That one formula might
> have two different sets of symbols that can express it is
> of course true, and uninteresting.
If I were you I'd not talk such a nonsense, lest that it would
become someone's signature - for laughing!
There's no such a thing as one formula having two different sets
of symbols, to start with. No one but you could even imagine what
that means!
> That a specific string of
> symbols might refer to one formula in one concrete syntax
> and a different formula in a different concrete syntax,
That and the below is really ... really a hopeless utterance.
Nam Nguyen
>
> What you are describing is relativity of concrete
> syntax, not relativity of truth.
Sigh! If you even bother to believe in some mathematical
relativity at all, why not believe in relativity of truth
instead of relativity of _concrete syntax_ which, by definition,
isn't relative?
Marshall
Well! Your prose is certainly getting more florid! That all
was very entertaining.
> Each of us is free to believe what's interesting or not.
Very true! Some people find it profound that "dog" spelled
backwards is "god." Others may be quite enthralled with
the fact that if you rearrange the order of the natural
numbers, you can make "3<2" a true statement.
However, neither of those things is at all profound.
> >>Would you believe that this meta statement:
> >>
> >>(*) There exist 2 prime numbers p1 p2 in which the
> >> statement "p1 is less than p2" is relativistic
> >> (i.e. subjective)
> >>
> >>is true ?
>
> Are you saying here that (*) is true?
It is essentially false. Any model of the natural numbers
is isomorphic to any other model of the natural numbers.
The only thing that can be done with the sorts of gyrations
you're attempting is to come up with a different carrier
set or a different successor order; neither of these
things affects whether a formula is true or not
in the resulting model.
You might want to read up on the phrase "up to
isomorphism." Not sure why I would say that,
since I cannot recall a single instance of you
even reading the wikipedia page for a concept
or term I recommend, but what the hell.
In short, you can change the names; you can
shuffle things around, but in the end you'll still
have something isomorphic to the standard model.
Marshall
Nam Nguyen
> On Dec 13, 11:13 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Marshall wrote:
>>
>> [...]
>
> Well! Your prose is certainly getting more florid! That all
> was very entertaining.
>
>
>> Each of us is free to believe what's interesting or not.
>
> Very true! Some people find it profound that "dog" spelled
> backwards is "god." Others may be quite enthralled with
> the fact that if you rearrange the order of the natural
> numbers, you can make "3<2" a true statement.
But, Marshall, who beside you is talking about "SSS0<SS0" being
true here? Why don't you read the conversation carefully!
> However, neither of those things is at all profound.
Only you uttered the word "profound" about phantom statement that
nobody else stated! You seem to have a kind of weird fixation on the
words "profound", "interesting", etc...
>
>
>> >>Would you believe that this meta statement:
>> >>
>> >>(*) There exist 2 prime numbers p1 p2 in which the
>> >> statement "p1 is less than p2" is relativistic
>> >> (i.e. subjective)
>> >>
>> >>is true ?
>>
>> Are you saying here that (*) is true?
>
> It is essentially false.
What does "essentially false" mean? Is that different from just
being false?
> Any model of the natural numbers
> is isomorphic to any other model of the natural numbers.
How does isomorphism among these different models invalidate
(*)? Do you have specific (technical) explanation? Or are you
just saying it?
> The only thing that can be done with the sorts of gyrations
> you're attempting is to come up with a different carrier
> set or a different successor order; neither of these
> things affects whether a formula is true or not
> in the resulting model.
What about the formulas that it's impossible to know whether or
not they are true in any resulting models?
>
> You might want to read up on the phrase "up to
> isomorphism." Not sure why I would say that,
> since I cannot recall a single instance of you
> even reading the wikipedia page for a concept
> or term I recommend, but what the hell.
You don't have proofs of your opponent's understanding or not
about "up to isomorphism", but why don't you concentrate in giving
a resounding yes or no (or I-don't-know) answer my questions
above and about (*), if you really understand what the thread
is about.
Just randomly keeping theorizing whether or not your opponent
knows about this or that doesn't make people understand *your*
arguments are.
>
> In short, you can change the names; you can
> shuffle things around, but in the end you'll still
> have something isomorphic to the standard model.
Again, how does this isomorphism invalidate (*) - specifically?
Marshall
> Marshall wrote:
>
> >> >>Would you believe that this meta statement:
>
> >> >>(*) There exist 2 prime numbers p1 p2 in which the
> >> >> statement "p1 is less than p2" is relativistic
> >> >> (i.e. subjective)
>
> >> >>is true ?
>
> >> Are you saying here that (*) is true?
>
> > It is essentially false.
>
> What does "essentially false" mean? Is that different from just
> being false?
The proposition is false.
I have tried to be very clear about what renaming can
and cannot accomplish, because you have been prone
in the past to argumentation along the lines of "what
if we rename true to "false"? It's not clear how much
of that you took in, though.
> > Any model of the natural numbers
> > is isomorphic to any other model of the natural numbers.
>
> How does isomorphism among these different models invalidate
> (*)?
> [...]
> What about the formulas that it's impossible to know whether or
> not they are true in any resulting models?
> [...]
> Again, how does this isomorphism invalidate (*) - specifically?
For all formulas in the language of basic arithmetic,
regardless of whether they are true or not, regardless
of whether their truth is knowable or not, they have
a definite truth value in the standard model. And
whatever a formula's truth value is in the standard
model, it will be the same in any model isomorphic
to the standard model. That what the word
"isomorphic" means.
Marshall
RussellE
> Infinitely many choices exist for a concrete syntax for FOL's
> abstract syntax; which choice we pick is uninteresting.
Computer scientists find it interesting.
> Of course the specific structure of that concrete syntax
> has to match the abstract syntax, just as the abstract
> syntax has to match the semantics; this does not change
> the fact that the choice of the concrete syntax is
> arbitrary and of no interest.
The choice of "concrete syntax" does make a difference
in which functions are "hard" and which ones are "easy".
Take addition as an example.
Creating an adder circuit depends on whcih binary codes
you assign to the natural numbers.
Some encodings allow much simpler addition circuits.
This is one reason most computers use two's complement notation.
http://en.wikipedia.org/wiki/Two's_complement
(two's complement notation actually makes subtraction easier.)