Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.
Dismiss

The set of natural numbers is not recursive

402 views
Skip to first unread message

Julio Di Egidio

unread,
Jun 20, 2021, 10:19:11 AM6/20/21
to
Let U be the universe of all numbers.

Predicates on numbers:

x in y in U x PU -> B := "in(x,y)"
x=y in U x U -> B := "eq(x,y)"

Basic definitions:

0_N in U := "zero_N()"
s_N(m) in U -> U := "succ_N(m)"

Natural numbers:

N := { n | n = 0_N \/ [
exists m . m in N /\ n = s_N(m) ] }

It's easy to see that, as a subset of the universe of numbers, N is not
a recursive set.

Conversely: The standard set of natural numbers is the set of those
natural numbers that are natural numbers...

Julio

Greg Cunt

unread,
Jun 20, 2021, 10:58:03 AM6/20/21
to
On Sunday, June 20, 2021 at 4:19:11 PM UTC+2, ju...@diegidio.name wrote:

> the set of those natural numbers that are natural numbers...

Hmmm... Are there any natural numbers which aren't natural numbers?

Dan Christensen

unread,
Jun 20, 2021, 11:51:17 AM6/20/21
to
Can you use this definition prove that there are natural numbers not equal to 0? How do you rule out 0 being its own successor?

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com


Mostowski Collapse

unread,
Jun 20, 2021, 12:27:22 PM6/20/21
to
A recursive set is a set where the set is recursively enumerable,
and the complement is recursively enumerable.

To the best of my knowledge N and N \ N = {}, are both recursively
enumerable. Or do you think they are not?

https://en.wikipedia.org/wiki/Computably_enumerable

https://en.wikipedia.org/wiki/Computable_set


Julio Di Egidio schrieb:

Mostowski Collapse

unread,
Jun 20, 2021, 12:31:41 PM6/20/21
to

From Wiki:
"A is a computable set if and only if A and
the complement of A are both c.e.."

{} is trivially c.e., since a finite set,
which is given in Peano numeral form, is c.e.,

N is trivially c.e., you can enumerate
it, in Peano numeral form:

0
s(0)
s(s(0))
s(s(s(0)))
Etc...



Mostowski Collapse schrieb:

Julio Di Egidio

unread,
Jun 20, 2021, 5:40:07 PM6/20/21
to
On 20/06/2021 16:19, Julio Di Egidio wrote:

> It's easy to see that, as a subset of the universe of numbers, N is not
> a recursive set.

That's the claim, and it does not depend on my specific definitions.
Claim: it's easy to see (unless one cannot even read) that, *as a
(possibly proper) subset of a universe of numbers*, N is *not* recursive.

> Conversely: The standard set of natural numbers is the set of those
> natural numbers that are natural numbers...

Maybe "conversely" should have been "by contrast", pardon my English.
By contrast, the natural numbers are "standardly" said to be
recursive... in the realm of the natural numbers, which is worse than
pointless: take any actual predicate nat/1 that is true iff the input is
a natural numbers, and see how recursive it is unless yo do always pass
a natural number! Recursive my ass... no?

Julio

Ross A. Finlayson

unread,
Jun 20, 2021, 6:31:00 PM6/20/21
to
Recursively ....

Haltingly halt in looping for example is for static analysis.

It results in much gains to estimate the recursive bounds....

That prime arithmetic solves membership test by remainder,
is that the number of primes and all their roots, in for example
the integer bounds: static analysis still provides great inroads
what that in the limits, such arithmetic of primes and roots is complete.

Here instead you are talking about the pathological "in the definition,
of recursive", that the direct sum starts like empty but is full.

It's a definition that after recursivity and computability,
there is a natural direct sum what is empty as you describe.

That's written out instead that "the rule is that it is full".

So, if you are to confront such roots in meaning what that
the naturals are compact, or so, that infinity is effectively large,
it might help keep in mind that for some the "current" or "received"
definition here has this addenda that of course admits such
full-ness for the compact and number theory as full-ness for
the "infinity empty sum", what here is either N or N+1, for
example, in an assignment, of either way the definition an ordinal.

Ben Bacarisse

unread,
Jun 20, 2021, 9:06:28 PM6/20/21
to
Mostowski Collapse <janb...@fastmail.fm> writes:

> From Wiki:
> "A is a computable set if and only if A and
> the complement of A are both c.e.."

But the definition states that the term applies to subsets of N
(i.e. the complement is taken relative to N). JDE is inventing the
notion of "recursive as a subset of X" and choosing X so as not to be
c.e. I'm not sure why he cares about this new meaning. The thread
title is deceptive (because this new notion is not explicitly
referenced) so maybe that's the point.

>> Julio Di Egidio schrieb:
<cut>
>>> It's easy to see that, as a subset of the universe of numbers, N is
>>> not a recursive set.

--
Ben.

Python

unread,
Jun 20, 2021, 9:27:14 PM6/20/21
to
It's been ages we've seen Julio Di Egidio pretending to have a point,
quite a very strong one actually. But nobody ever succeeded in
determining even what he was talking about.


Graham Cooper

unread,
Jun 20, 2021, 10:16:57 PM6/20/21
to
On Monday, June 21, 2021 at 12:19:11 AM UTC+10, ju...@diegidio.name wrote:
> Let U be the universe of all numbers.
>
> Predicates on numbers:
>
> x in y in U x PU -> B := "in(x,y)"
> x=y in U x U -> B := "eq(x,y)"
>
> Basic definitions:
>
> 0_N in U := "zero_N()"
> s_N(m) in U -> U := "succ_N(m)"
>
> Natural numbers:
>
> N := { n | n = 0_N \/ [
> exists m . m in N /\ n = s_N(m) ] }
>
> It's easy to see that, as a subset of the universe of numbers, N is not
> a recursive set.

bwahaha

m in N ^ n = s(m)

just use

e( 0 , n ).
e( s(X) , n ) :-
<-
e(X , n)


how you infer it is NOT recursive ?






> Conversely: The standard set of natural numbers is the set of those
> natural numbers that are natural numbers... ~ Julio (sci.logic)

SIG MATERIAL! whatever it means






Mostowski Collapse

unread,
Jun 21, 2021, 1:46:04 AM6/21/21
to
He uses U, but if he has mathematical induction:

0 e X & forall n (n e X => s(n) e X) => forall n (n e X)

Then we can derive forall n (n e N). Which amounts to U ⊆ N.
To have possibly an universe U different from N, he would
need to state mathematical induction relativized:

0 e X & forall n (n e N => n e X => s(n) e X) => forall n (n e N => n e X)

That is what omega ω inside set theory can do. But
to show that V \ ω is non empty, one needs more set
theory, not only the signature like:

x in y in U x PU -> B := "in(x,y)"
x=y in U x U -> B := "eq(x,y)"

Without further axioms U \ N non empty isn't decidable,
neither U \ N non empty nor U \ N empty follows.

Julio Di Egidio

unread,
Jun 21, 2021, 3:21:27 AM6/21/21
to
On 21/06/2021 03:06, Ben Bacarisse wrote:
> Mostowski Collapse <janb...@fastmail.fm> writes:
>>> Julio Di Egidio schrieb:
> <cut>
>>>> It's easy to see that, as a subset of the universe of numbers, N is
>>>> not a recursive set.>> From Wiki:
>> "A is a computable set if and only if A and
>> the complement of A are both c.e.."
>
> But the definition states that the term applies to subsets of N
> (i.e. the complement is taken relative to N). JDE is inventing the
> notion of "recursive as a subset of X" and choosing X so as not to be
> c.e.

I am inventing nothing, you guys are simply too stupid to even read
English. Later in that same article: "The entire set of natural numbers
is computable." <https://en.wikipedia.org/wiki/Computable_set>

And, to reiterate, *of course* that is correct, since it is in terms of
subsets of N! Then my qualm is that it is worse than vacuous... but we
won't get there if you cannot even parse this far.

> I'm not sure why he cares about this new meaning. The thread
> title is deceptive (because this new notion is not explicitly
> referenced) so maybe that's the point.

It must suck to by you(s)... I mean, just please cut that crap.

Julio

Julio Di Egidio

unread,
Jun 21, 2021, 3:43:26 AM6/21/21
to
On 21/06/2021 00:30, Ross A. Finlayson wrote:
> On Sunday, June 20, 2021 at 2:40:07 PM UTC-7, ju...@diegidio.name wrote:
<snip>
> > By contrast, the natural numbers are "standardly" said to be
> > recursive... in the realm of the natural numbers, which is worse than
> > pointless: take any actual predicate nat/1 that is true iff the input is
> > a natural numbers, and see how recursive it is unless yo do always pass
> > a natural number! Recursive my ass... no?
>
> Recursively ....
>
> Haltingly halt in looping for example is for static analysis.

From the book of things to defend no matter what? In the general case,
it *fails* decidability. Which is pretty much my point.

Julio

Graham Cooper

unread,
Jun 21, 2021, 10:06:33 AM6/21/21
to
On Monday, June 21, 2021 at 5:21:27 PM UTC+10, ju...@diegidio.name wrote:
> On 21/06/2021 03:06, Ben Bacarisse wrote:
> > Mostowski Collapse <janb...@fastmail.fm> writes:
> >>> Julio Di Egidio schrieb:
> > <cut>
> >>>> It's easy to see that, as a subset of the universe of numbers, N is
> >>>> not a recursive set.>> From Wiki:
> >> "A is a computable set if and only if A and
> >> the complement of A are both c.e.."
> >
> > But the definition states that the term applies to subsets of N
> > (i.e. the complement is taken relative to N). JDE is inventing the
> > notion of "recursive as a subset of X" and choosing X so as not to be
> > c.e.
> I am inventing nothing, you guys are simply too stupid to even read
> English. Later in that same article: "The entire set of natural numbers
> is computable." <https://en.wikipedia.org/wiki/Computable_set>

the individuals of the entire set are computable

the SET itself isnt computable by standard definition





>
> And, to reiterate, *of course* that is correct, since it is in terms of
> subsets of N! Then my qualm is that it is worse than vacuous... but we
> won't get there if you cannot even parse this far.


its as vacuous as

A(L) {0,1}eL_n_d E(m) A(r) m=/=L_r

or as DAN puts it... infinite lists of ANY SIZE (finite or infinite) have a missing row

but you DONT GET THAT FAR

Mostowski Collapse

unread,
Jun 21, 2021, 12:11:42 PM6/21/21
to
Graham Cooper haluzinated:
"the SET itself isnt computable by standard definition"

Read the wiki:
- The entire set of natural numbers is computable.
https://en.wikipedia.org/wiki/Computable_set

Nobody uses WM nitpicking language in real world,
when its anyway clear what is meant.

Ross A. Finlayson

unread,
Sep 30, 2022, 12:26:02 PM9/30/22
to
"Starts like empty but is full, ...".

"There are a lot of hangnails, ..., in usual deft formalism."

Jeffrey Rubard

unread,
Sep 30, 2022, 5:32:28 PM9/30/22
to
It's... iterative?

Jeffrey Rubard

unread,
Oct 1, 2022, 6:41:11 PM10/1/22
to
Because of the "successor function"?
"Tell me about it."
Maybe you could read an introductory text on number theory or set theory instead?

Ross A. Finlayson

unread,
Oct 2, 2022, 7:33:38 PM10/2/22
to
Sure, I got nothing for you.

Jeffrey Rubard

unread,
Oct 2, 2022, 7:58:43 PM10/2/22
to
Richard Dedekind's *Was sind und was sollen die Zahlen?* is the "locus classicus" of the idea.

Ross A. Finlayson

unread,
Oct 2, 2022, 10:49:32 PM10/2/22
to
Was sind and was zollen die Zahlen?

Yeah, sure.

Dedekind completeness here though is only one part of continuity,
there is also after Eudoxus same as Dedekind same as Cauchy,
completeness, pretty much most all what's acribed to Dedekind.

Yeah, knowing nothing, though, it's been many yeats since I read
Was sind and was zollen die Zahlen, probably a translation.

Many yeats....

Actually, I read an introductory text and and number theory, set theory.

Eudoxus/Dedekind/Cauchy is all one completeness, Dedekind's.

It's called "least upper bound".

The rule is that it is full, ....

Ross A. Finlayson

unread,
Oct 2, 2022, 10:52:52 PM10/2/22
to
Otherwise the rule would be that it is empty, ....

Then there's "half-full".

Ross A. Finlayson

unread,
Oct 2, 2022, 10:59:45 PM10/2/22
to
The axiomless natural deduction is all perfect and I really read a lot into it.

It's the ex falso nihilum not the ex falso quodlibet.

Out of falsehood is nothing, not "out of falsehood is lies".

Sure, I have nothing, for you. It's called axiomless natural deduftion,
and somehow perfect in every way.

I don't know what you get out of "nothing" but here it's a pretty big deal.

Pretty much all I need for everything.



They didn't already have one, ....



Jeffrey Rubard

unread,
Oct 3, 2022, 9:54:50 PM10/3/22
to
Yeah, uh...

> Yeah, knowing nothing, though, it's been many yeats since I read
> Was sind and was zollen die Zahlen, probably a translation.
>
> Many yeats....

You found some others besides William Butler?
"Yes."
I think you might be a bit of a Philistine, actually.

>
> Actually, I read an introductory text and and number theory, set theory.
>

Yeah, me too. What a big deal for other people! The only worse thing is
going "on and on" when you don't even have the basics, like you seem to do
quite a bit.

Ross A. Finlayson

unread,
Oct 4, 2022, 12:11:47 AM10/4/22
to
How about something from a series edited by RKT Hsieh?

You think I don't have "the basics"? Wow, and I thoght it was
all 0's and 1's down here and true or false up here.



So, is there anything to care out of Was sind and was zollen die Zahlen,
except least-upper-bound?

By care I mean, you know, originated there.


Really these books edited by RKT Hsieh have a lot going on.

Makes Dedekind look like a finger-counter.

Not that there's anything wrong with that, ....


So, I imagine though you would have, ...,
"least-upper-bound is that the rule is that it is full", ...,
the ordered field, which otherwise isn't.

I'm not here to bring you down, though I have no qualms
breaking logic down, and I don't know about you,
but axiomless natural deduction is pretty much the floor.


So, if you detail the axiomatization of the complete ordered field,
which includes Dedekind "axiomatizing", least-upper-bound, and
what, Hilbert? Borel? axiomatizing "measure 1.0", here I'm not sure exactly
who has ascribed originating "measure 1.0", yet another clumsy axiom
from (... those) unable to reason from first principles.


Really though if I want to learn about waves or something there's lots
from series edited by RKT Hsieh and like Maugin and so on, interesting things.


If you're really interested in demonstrating the elementary,
there are all the other theories somehow all one theory.

How could that be?

So, just to ... reiterate: least-upper-bound is an axiom "the rule is
that it is full", and it's not so different from axioms like infinite sets
"ditto" and other notions of completeness "ditto" and in at least some
theories after monism, it's provided as an inference without needing
to be axiomatized, where axioms are stipulations, and one needn't be an ass.



For example from iota-values a model of reals, least-upper-bound
and measure 1.0 fall right out. (And it's very simple.)

Didn't somebody already, you know, burning-bush you that or something?

Guess not.





Ross A. Finlayson

unread,
Oct 4, 2022, 12:27:05 AM10/4/22
to
This "Borel vs. Combinatorics" is pretty great,
"almost everywhere", "almost nowhere", "half".

See, the rule is that it "meets in the middle".

Ross A. Finlayson

unread,
Oct 4, 2022, 12:52:21 AM10/4/22
to
Actually I read a fundamental set theory, number theory, ....

It reads right off, ....

Ross A. Finlayson

unread,
Oct 4, 2022, 1:35:54 AM10/4/22
to
I prefer Frege to Dedekind, and Galileo defined "infinite sets" just fine.

Jeffrey Rubard

unread,
Oct 6, 2022, 6:42:47 PM10/6/22
to
That's... not how that works. "Mathematics is mathematics."

Mostowski Collapse

unread,
Oct 7, 2022, 8:00:14 PM10/7/22
to
ju...@diegidio.name schrieb am Sonntag, 20. Juni 2021 um 16:19:11 UTC+2:
> Let U be the universe of all numbers.

Thats something for wonky man. He also makes a recurring
error to start modelling the universe of discourse, as a
set inside the universe of disourse.

As soon as you do that, it seems you have elements that
then jump out of your universe of discourse. Assume you want
to have a unverse of discourse of natural numbers, but

you make the error to also introduce sets of natural numbers,
etc.. etc.., i.e. you go a little bit too far what you otherwise
allow in your universe of discourse. Eh volia, you

open pandoras box:

/* Previous Result http://www.dcproof.com/UniversalSet.htm */
1 ALL(s):[Set(s) => EXIST(a):~a ε s]
Axiom

2 Set(nat)
Axiom

3 Set(nat) => EXIST(a):~a ε nat
U Spec, 1

4 EXIST(a):~a ε nat
Detach, 3, 2

Oopsi, what is this thing outside of N?

Whats is the classical way, for example in FOL, to not
be subject of this paradox, the wonky man paradox?

Julio Di Egidio

unread,
Oct 8, 2022, 8:20:32 AM10/8/22
to
On Saturday, 8 October 2022 at 02:00:14 UTC+2, Mostowski Collapse wrote:
> ju...@diegidio.name schrieb am Sonntag, 20. Juni 2021 um 16:19:11 UTC+2:
> > Let U be the universe of all numbers.
>
> Thats something for wonky man. He also makes a recurring
> error to start modelling the universe of discourse, as a
> set inside the universe of disourse.

What are you even talking about?! Indeed, he may have
"logical" problems, but the one who keeps throwing and
thriving on just word salad is you.

> open pandoras box:
>
> /* Previous Result http://www.dcproof.com/UniversalSet.htm */
> 1 ALL(s):[Set(s) => EXIST(a):~a ε s]
> Axiom

So, if I read/unpack it at face value:

Firstly, to the point you raise above, the "universe of
discourse" here is more than just sets, no doubt on that,
or we would not write "ALL(s)-such-that-s-is-a-set" and
would just write "ALL(x)". Rather, in that universe, there
must be things called sets, thanks the predicate Set(x).

That said, the axiom is stating that "for every set 's', there
exists an *object* 'a' (i.e. not necessarily a set, as stated!)
such that 'a' not in 's'".

Now, should sets here only contain other sets (as they
standardly do, and as I guess the intention here was),
via some other assumption/axiom not shown here, then
the axiom above is poorly written to begin with, as an 'a'
not constrained to be a set could not appear on the LHS
of that epsilon to begin with.

So, let's rather consider the two cases:

1) if 'a' is constrained to be a set, the axiom ends up
stating the indicated "there is no set of all sets"... or
does it?? In a well-founded theory of sets the axiom
is indeed and necessarily true since unique candidate
for such 'a' would be the set of all sets itself, but then
*it* is no element of any set not even itself, so no cigar;
with non-well founded sets I suppose it becomes
possible to have a set of all sets, and the above is a
"contingent" axiom of the kind "thou shall not have a
such thing" and the consequences come down
the line...

2) if 'a' instead is an arbitrary element from the domain
of discourse, the axiom is vacuously true if there actually
exist objects that are not sets in the universe; otherwise
as above...

(I have not given it much detailed thinking, better double-
check all of the above.)

> 2 Set(nat)
> Axiom
>
> 3 Set(nat) => EXIST(a):~a ε nat
> U Spec, 1
>
> 4 EXIST(a):~a ε nat
> Detach, 3, 2
>
> Oopsi, what is this thing outside of N?
>
> Whats is the classical way, for example in FOL, to not
> be subject of this paradox, the wonky man paradox?

To begin with, don't write gibberish. Indeed, what are you
even trying to say: that 'nat' is "a universe"?? Of course it
is *not*, i.e. at least as long as there exists in the universe
things, even sets!, that are not in fact natural numbers.

Julio

Fritz Feldhase

unread,
Oct 8, 2022, 10:38:58 AM10/8/22
to
On Saturday, October 8, 2022 at 2:00:14 AM UTC+2, Mostowski Collapse wrote:
> ju...@diegidio.name schrieb am Sonntag, 20. Juni 2021 um 16:19:11 UTC+2:
> >
> > Let U be the universe of all numbers.
> >
> Thats something for wonky man. He also [...]

Of course we can restrict your universe of discourse to, say, "the natural numbers".

This way, you can just write, say,

An(s(n) =/= 0) ,

etc.

> As soon as you do that, it seems you have elements that
> then jump out of your universe of discourse.

This will only happen if you allow for some axioms in your "system" which make certain existence claims (for example concernig the existence of sets...)

> Assume you want to have a unverse of discourse of natural numbers,
> but you make the error to also introduce sets of natural numbers,
> etc.. etc..,

Right.

> /* Previous Result http://www.dcproof.com/UniversalSet.htm */
> 1 ALL(s):[Set(s) => EXIST(a):~a ε s]
> Axiom

No problem if in addition we have: ALL(s):[~Set(s)] (i. e. if we are not interested in sets at all).

> 2 Set(nat)
> Axiom

Ooops...

In addition with the axiom

An(n e nat)

this would lead to a contradiction. Now (in this context) we don't want to talk about a set /nat/ of natural numbers. After all, we started with (mentally) "restricting" the universe of discourse to "the natural numbers" (and most likely the set of natural numbers isn't a natural number, I'd say).

> Oopsi, what is this thing outside of N?

What is N? :-P

> Whats is the classical way, for example in FOL, to not
> be subject of this paradox [...]

Beware of the sets! :-)

Hmmm... Doesn't this approach just lead to the system usually called PA? A quote from some internet document:

"One simple solution is to design a “first-order” theory of N in which the universe is supposed
to be N and the underlying language is [0, s; =]."

This was done on pages 49-50, and the result is a complete theory Th(s) which can be completely axiomatized. However this theory
cannot formulate much of interest, because + and · cannot be defined in this language.

Thus to formulate our theory PA we extend this simple language by adding + and · to obtain the language LA = [0, s, +, ·; =]."

See: https://www.cs.toronto.edu/~sacook/csc438h/notes/page96.pdf

Julio Di Egidio

unread,
Oct 8, 2022, 1:54:48 PM10/8/22
to
On Saturday, 8 October 2022 at 16:38:58 UTC+2, Fritz Feldhase wrote:
> On Saturday, October 8, 2022 at 2:00:14 AM UTC+2, Mostowski Collapse wrote:
> > ju...@diegidio.name schrieb am Sonntag, 20. Juni 2021 um 16:19:11 UTC+2:
<snip>
> In addition with the axiom
>
> An(n e nat)
>
> After all, we started with (mentally) "restricting"

Yeah, "mentally" is the right word there.

> > Oopsi, what is this thing outside of N?
>
> What is N? :-P
>
> > Whats is the classical way, for example in FOL, to not
> > be subject of this paradox [...]
>
> Beware of the sets! :-)
>
> Hmmm... Doesn't this approach just lead to the system
> usually called PA?

It's just upside-down on the whole line, all the more so in
the context of this thread, thank you as usual. Indeed, PA
is, to begin with, a theory of sets, and just among those
sets some encode natural numbers. So, if anything, one
should add *restrictions* in order to talk exclusively about
natural numbers, i.e. the opposite of a need for existence
axioms.

> A quote from some internet document:
>
> "One simple solution is to design a “first-order” theory
> of N in which the universe is supposed
> to be N and the underlying language is [0, s; =]."

A solution to what?? A rhetorical question, you other
spamming idiot...

*Plonk*

Julio

Fritz Feldhase

unread,
Oct 8, 2022, 2:37:15 PM10/8/22
to
On Saturday, October 8, 2022 at 7:54:48 PM UTC+2, ju...@diegidio.name wrote:
>
> Indeed, PA is, to begin with, a theory of sets, and

PA is w h a t?!

Julio Di Egidio

unread,
Oct 8, 2022, 3:40:56 PM10/8/22
to
Eh, I was thinking sets, e.g. what one does in ZF: regardless,
my objection to you remains, that existence axioms are for
"structuring/singling out things" in a domain of discourse,
but are not exclusive of anything re what is possible by the
underlying theory. You'd need *non*-existence axioms for
that. (Second order theories can be "categorical", but that's
another story: and even there, what is in the theory is not
simply and only what one explicitly gives a name to.)

And in all that, the "irony", you as JB are essentially making
the same mistake as Dan's, conflating "(the!) domain of
discourse" with the existence of specific classes/collections
and what a predicate, say nat(x), even means: overall
essentially categorical errors. Which is not surprising
anyway, since asserting that the set of natural numbers is
recursive... re a "universe" that is the set of natural numbers
itself is exactly along the same line of nonsense.

And with that I close the circle re this thread: additional
pollution I'd wish you and co. just kept where it belongs.

EOD.

Julio

Ross A. Finlayson

unread,
Oct 8, 2022, 5:32:36 PM10/8/22
to
Hm.

Here there's splitting universal quantifier

for-any for-every for-each for-all

then having "exists" quantifier and "exists-unique".

This way quantifier disambiguation is simply organized
under conventions, of definition,
of any, each, every, all,
where, about those mean the same,
really between themselves they establish quantifier disambiguation.

Then that's also one quantifer and infinite quantifiers in front
of one variable and infinite variables, then, those in terms.

A model of which is a constant. (Which is among reasons why
bounded and standard and completed and nonstandard,
models exist, if not a standard generic model.)

Mostowski Collapse

unread,
Oct 8, 2022, 5:49:25 PM10/8/22
to
The theorem:

ALL(s):[Set(s) => EXIST(a):~a ε s]

Is in contradiction to what we expect from U:

Set(U)

ALL(a):[a e U]

Mostowski Collapse

unread,
Oct 8, 2022, 5:57:05 PM10/8/22
to
Even DC Spoild itself can derive the contradiction:

1 ALL(s):[Set(s) => EXIST(a):~a ε s]
Axiom

2 Set(u)
Axiom

3 ALL(a):a ε u
Axiom

4 Set(u) => EXIST(a):~a ε u
U Spec, 1

5 EXIST(a):~a ε u
Detach, 4, 2

6 ~ALL(a):~~a ε u
Quant, 5

7 ~ALL(a):a ε u
Rem DNeg, 6

8 ALL(a):a ε u & ~ALL(a):a ε u
Join, 3, 7

Mostowski Collapse

unread,
Oct 8, 2022, 7:19:40 PM10/8/22
to
Dan Christensen is currently banging his head in the
other direction. Although Set(U) and ALL(a):[a e U]
leads to a contradiction, on the other hand when we

use predicate symbols, this here isn't contradictory per se:

ALL(a):U(a)

It is even not related to empty domain issues. Its just
the fact that a predicate can be "full". It also
causes a counter model here:

∃x(Dx→Q) is invalid.
https://www.umsu.de/trees/#~7x%28D%28x%29~5Q%29

Another way to say "full", is to write ∀yD(y),
which is what makes the Drinker Paradox work,
namely that we don't have an arbitrary Q,

but that we have ∀yD(y). In the Drinker Paradox
you can translate "full" into "everybody
is drinking (in this round)".

Julio Di Egidio

unread,
Oct 9, 2022, 7:22:33 AM10/9/22
to
On Saturday, 8 October 2022 at 23:57:05 UTC+2, Mostowski Collapse wrote:

> Even DC Spoild itself can derive the contradiction:

What derivation for what contradiction??

> 1 ALL(s):[Set(s) => EXIST(a):~a ε s]
> Axiom

"Axiom: There is no universal set (i.e. a set of all things
in the universe of discourse)."

> 2 Set(u)
> Axiom
>
> 3 ALL(a):a ε u
> Axiom

"Axiom: There is a universal set, call it u."

And then I fail to see what is interesting there.

Julio

Mostowski Collapse

unread,
Oct 9, 2022, 7:33:49 AM10/9/22
to
Deriving Q & ~Q for some formula Q, like here:

8 ALL(a):a ε u & ~ALL(a):a ε u
Join, 3, 7

where Q = ALL(a):a ε u,

is usually considered a contradiction.

Julio Di Egidio

unread,
Oct 9, 2022, 7:44:37 AM10/9/22
to
On Sunday, 9 October 2022 at 13:33:49 UTC+2, Mostowski Collapse wrote:
> Deriving Q & ~Q for some formula Q, like here:
> 8 ALL(a):a ε u & ~ALL(a):a ε u
> Join, 3, 7
> where Q = ALL(a):a ε u,
> is usually considered a contradiction.

Are you really missing the point?? Your 1 already implies
~(2-3), so *essentially* *you* have assumed ~Q *and* Q,
in that order, you blistering idiot, and the only interesting
thing about that is that at least DCProof is not as buggy
as you are!

Ge the fuck out of here...

*Plonk*

Julio

Mostowski Collapse

unread,
Oct 9, 2022, 7:44:40 AM10/9/22
to
Basically when you derive Q & ~Q, you
state that Q & ~Q is a tautology, which it is not:

Q Q & ~Q
0 0
1 0

Its even not sometimes true, its always false!
As soon as Q & ~Q enters your system,

you can derive anything. Even DC Proof can do it:

1 Q & ~Q
Axiom

2 ~P
Premise

3 Q
Split, 1

4 ~Q
Split, 1

5 Q & ~Q
Join, 3, 4

6 ~~P
Conclusion, 2

7 P
Rem DNeg, 6

This is called ECQ, ex contradictione quodlibet, if you use
ff instead of Q & ~Q, you get EFQ, ex falso quodlibet, but
often ff and Q & ~Q are used interchangeably:

'Ex Falso Quodlibet' is the mediaeval name for the rule of inference
which allows that from a contradiction you may deduce anything whatsoever.

(Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or
ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]')
https://en.wikipedia.org/wiki/Principle_of_explosion#Symbolic_representation

Ross A. Finlayson

unread,
Oct 9, 2022, 11:52:22 AM10/9/22
to
Excluded middle is just a split.

Ross A. Finlayson

unread,
Oct 9, 2022, 11:55:56 AM10/9/22
to
That's what you get for getting led along.

Ross A. Finlayson

unread,
Oct 9, 2022, 12:05:58 PM10/9/22
to
Here an example is "division by zero". All these people starting with zero and numbers,
then adding division, leaving out zero instead of no zero, these are clumsy and cumbersome.

Not contradicting each other while still have mutually compatible interpretations of views,
what result abstractly the value of fairness where it's the only principle, that all the
definitions are composed into one definition, besides that inference exists.

One of the other here for example "numbers with zero" and "numbers and division",
entirely are differently varied smaller collections of direct resulting inferences,
that only one or the other is eventually exhausted in any sense of completion.

DC Proof is unsafe, it's stupid. What it is.

Of course I mostly write direct definition, I expect also all these
words compose, also.

Rather, all the symbolic exhaustion to completion, closures and
such and limits of unity, are also fields, it results for composable
inference, here for a compilation unit that there's lexical scope
and definiton.

Which is only "true template" and "garbage in, garbage out".

Anyways the Wiki has been helping understand this "circularity
or ..., or ... ", apologizing, sorting out systems oif inference in some basic objects.

Syllogism is all sorting, and only what results stable sorting
under all alternatives, otherwise what's called "mutual" inference.

Julio Di Egidio

unread,
Oct 10, 2022, 5:30:45 AM10/10/22
to
On Sunday, 9 October 2022 at 18:05:58 UTC+2, Ross A. Finlayson wrote:

> DC Proof is unsafe, it's stupid. What it is.

No, you are being stupid, and the whole gang here.

> Of course I mostly write direct definition, I expect
> also all these words compose, also.

(IIRC) DCP gives you a formal *language*, a first-order
one, including an engine to just validate a proof's steps:
what you write with it, your own *theory*, with its axioms,
definitions and the theorems you prove, that is up to you!

So "unsafe" in the sense that it won't find out for you if
*your* set of axioms is self-contradictory?? You tell me
one single system on the market that is able to do that
and we have a winner for the halting problem...! Moron.

So, DCP is a simple system and to today I still have to see
anybody come up with an actual technical problem with it.
Altogether different issue is how Dan uses it, and in that
he is indeed not any better than you are, essentially and
systematically lying, "lying with logic" here.

My humble advice: stop spamming this and every thread
(do you realize that this is all just off-topic for this thread
and I am replying only because I am a friend of yours and
you seem in serious pain?), rather try and get a grip on
yourself to begin with. You and the whole gang...

And that's it. EOD. Please. God. Help us all.

Julio

Mostowski Collapse

unread,
Oct 10, 2022, 5:49:32 AM10/10/22
to

LEM doesn't lead to ex falso quodlibet.
For example intuitionistic logic has no LEM,

but it still has ex falso quodlibet. How do
you explain this Mr. Ross A. Finlayson?

Answer: Intuitionistic logic has no LEM,
but it has still LNC:

~(p & ~p)
https://en.wikipedia.org/wiki/Law_of_noncontradiction

Have Fun Rossy Boy!

Mostowski Collapse

unread,
Oct 10, 2022, 6:01:55 AM10/10/22
to

But I guess you need something stronger than
LNC itself, to get ex falso quodlibet. For example
this here is provable in minimal logic:

fol(g4m)>
------------- Ax
p,bot --> bot
---------------- L0->1
p->bot,p --> bot
------------------- R->
p --> (p->bot)->bot
--------------------- R->
--> p->(p->bot)->bot
------------------------ Ltop
top --> p->(p->bot)->bot

But this here is not provable in minimal logic:

fol(g4m)>
Fail to prove top --> p->(p->bot)->q

On the other hand it is provable in intuitionistic logic
and classical logic (because of the Lbot rule):

fol(g4i)>
fol(g4c)>
----------- Lbot
p,bot --> q
-------------- L0->1
p->bot,p --> q
----------------- R->
p --> (p->bot)->q
------------------- R->
--> p->(p->bot)->q
---------------------- Ltop
top --> p->(p->bot)->q

Was fiddling with this prover:
https://www.vidal-rosset.net/g4-prover/

Ross A. Finlayson

unread,
Oct 10, 2022, 12:19:11 PM10/10/22
to
No, anything that resolves to inconsistent is replaced with a template
of "Liar" and erased, ex falso nihilum.

Its points that result it wrong remain in a box,
but, then it's an exercise "that was wrong".

Jeffrey Rubard

unread,
Oct 11, 2022, 3:40:45 PM10/11/22
to
That is... masterful, simply masterful.
(People make fun of me when I write things like that, you know.)

Ross A. Finlayson

unread,
Oct 12, 2022, 11:50:11 AM10/12/22
to
Better than being, ..., "WRONG".

Good luck then and while snide snark is a usual sort of defense mechanism,
mathematical proof is enduring.


Mostowski Collapse

unread,
Oct 12, 2022, 5:13:59 PM10/12/22
to
In type theory everything could be different.
For example a sentence like:

ALL(x):U(x)

Cannot be formed. Only a sentence:

ALL(x_i):U(x_i)

Where _i refers to some type. So predicates
would be limited by the type of their arguments,

openening the door to Dan Christensens aka
wonky mans dream, of a set theory that is more

potent than predicates.

Jeffrey Rubard

unread,
Oct 12, 2022, 5:28:22 PM10/12/22
to
Dopey sophistry, isn't it?

Mostowski Collapse

unread,
Oct 13, 2022, 11:42:57 AM10/13/22
to

Russell's own answer to the puzzle came in the form of a "theory of types."
https://www.scientificamerican.com/article/what-is-russells-paradox

“Appendix B: The Doctrine of Types” of Russell 1903.
https://plato.stanford.edu/entries/type-theory/#ParaRussTypeTheo

Later:

Russell thought it necessary to introduce another kind of hierarchy,
the so-called “ramified hierarchy” Chapter 10 of Russell 1959
https://plato.stanford.edu/entries/type-theory/#RamiHierImprPrin

And then they wanted to undo it:

Because of these problems, Russell and Whitehead introduced in
the first edition of Principia Mathematica the following reducibility axiom:
the hierarchy of predicates, first-order, second-order, etc., collapses at level 1.

LoL

Mostowski Collapse

unread,
Oct 13, 2022, 11:47:44 AM10/13/22
to
Yet another paper by Russell and a contender to set theory.
Type theory being an alternative to set theory:

Type theory can be used as a foundation for mathematics, and
indeed, it was presented as such by Russell in his 1908 paper,
which appeared the same year as Zermelo’s paper, presenting
set theory as a foundation for mathematics.

Russell, 1908, “Mathematical logic as based on the theory of types,”
American Journal of Mathematics, 30: 222–262.

Ross A. Finlayson

unread,
Oct 14, 2022, 1:26:39 PM10/14/22
to
You'll notice ramified and stratified kind of become
interchangeable, in terms of what's first order (or lesser
order), Russell's type theory with ramified and stratified
types.

In set theory, ....

Mostowski Collapse

unread,
Oct 14, 2022, 8:24:39 PM10/14/22
to
Wonly man aka Dan Christen would be very much
helped if he could think in types. Types are much
eleganter than his no no and yes yes set nonsense.

Types also mirror much more what mathematicians
do in their text. They often introduce conventions like:

- n, m, etc.. natural numbers
- p, q, etc.. rational numbers
- etc..

So whats the meaning of this formula,
its even a theorem:

∀q ∃n ∃m (q = n/m)

Mostowski Collapse

unread,
Oct 14, 2022, 8:26:24 PM10/14/22
to
We can even apply this idea further, like for example:

- x, y, etc.. guests in the pub
- X, Y, etc.. sets of guests

So what does the Drinker Paradox says:

∃x (Dx -> ∀yDy)

Ross A. Finlayson

unread,
Oct 15, 2022, 12:23:00 AM10/15/22
to
Only one set of natural numbers is not recursive ....

There is one, though.

Think of rationals and irrationals as complements,
and for example integers and negative integers as complements.

Or, for example, integers and non-integer half-integers.

Types, ....

Mostowski Collapse

unread,
Oct 15, 2022, 7:08:55 AM10/15/22
to
Types would really salvage Dan Christensen aka Wonky
Man, from many of his mental deseases. For example
we could distinguish:

a) Inhabitated Types: Types that are never empty
b) Uninhabitated Types: Types that can be possibly empty.

Now he thinks the world needs a tool like DC Proof,
which allegedly works with an Uninhabitated
Domain of Discourse.

Whats his blind spot? Tools with a Inhabitated Domain
can also simulate Uninhabitated Domains, if they
have predicates that can be Uninhabtated.

LoL

Mostowski Collapse

unread,
Oct 15, 2022, 7:38:43 AM10/15/22
to
If I am not totally mistaken this feature
of FOL, to be able to model empty domains,
is even explained here:

Logic Lectures: Gödel's Basic Logic Course at Notre Dame
https://arxiv.org/abs/1705.02601

The Drinker Paradox is not the first formula
which depends on some existential import.
Many formulas appear with such a demand,

when one tries to formalize for example:

"The origin of the square can be traced back to
Aristotle's tractate On Interpretation."
https://en.wikipedia.org/wiki/Square_of_opposition#The_problem_of_existential_import

One can proceed as follows, as FOL does:

"It is ambiguous whether or not a universal statement of the
form All A is B is to be considered as true, false, or even meaningless
if there are no As. If it is considered as false in such cases, then
the statement All A is B has existential import with respect to A."
https://en.wikipedia.org/wiki/Syllogism#Existential_import

Mostowski Collapse

unread,
Oct 15, 2022, 7:49:54 AM10/15/22
to
The Drinker Paradox is maybe special since it uses none
of the forms A, E, I or O. Namely the root formula is not prima
facie of any of the forms, modern variants, and can also

hardly be braught into this form:

All A is B ALL(x):[A(x)=>B(x)]
No A is B ALL(x):[A(x)=>~B(x)]
Some A is B EXIST(x):[A(x)&B(x)]
Some A is not B EXIST(x):[A(x)&~B(x)]

Among the Square of Opposition, even if somebody
agrees on the meaing of ALL(x):[A(x)=>...] and
EXIST(x):[A(x)&...] for an empty A, if this is done

in the usual modern way à la FOL, then some relations
between A, E, I or O still need an extra non-empty
statement. See here:

case(19, darapti, (m a p, m a s, n m -: s i p)).
case(20, felapton, (m e p, m a s, n m -: s o p)).
case(25, bramantip, (p a m, m a s, n p -: s i p)).
case(28, fesapo, (p e m, m a s, n m -: s o p)).

n X is the non-empty statement. For more details
see for example here, case(19) and case(20):

1.2.3 Third Figure
Aristotle’s Assertoric Syllogistic
Angeliki Koutsoukou-Argyraki
December 14, 2021
https://www.isa-afp.org/browser_info/current/AFP/Aristotles_Assertoric_Syllogistic/document.pdf

You can use Wolfgang Schwartz tree tool generate to
find counter models when the non-empty statement
is missing. Yes or No? So the antique had already

Drinker Paradoxes that had dependency on a non-empty
side condition. But they hardly said it needs the
Russel Paradox. Thats Dan Christensen nuthead stuff.

LMAO!

Jeffrey Rubard

unread,
Oct 15, 2022, 8:01:22 PM10/15/22
to
Ooh, "drinker paradoxes", etc.

Mostowski Collapse

unread,
Oct 16, 2022, 6:04:44 PM10/16/22
to
It will take Wonky Man aka Dan Christensen a
100 years to understand his own fallacy in
the drinker paradox. Sets are not individuals,

they don't drink beer, they don't visit a pub.
Bending the drinker paradox from:

> - x, y, etc.. guests in the pub

To this here, and invoking the Russell Paradox,
isn't some smart move:

- X, Y, etc.. sets of guests

Mostowski Collapse schrieb am Samstag, 15. Oktober 2022 um 02:26:24 UTC+2:
> We can even apply this idea further, like for example:
>
> - x, y, etc.. guests in the pub
> - X, Y, etc.. sets of guests
>
> So what does the Drinker Paradox says:
>
> ∃x(Dx -> ∀yDy)

Mostowski Collapse

unread,
Oct 16, 2022, 6:11:08 PM10/16/22
to
Dan Christensen proves, and doesn't think any further:

ALL(s):[Set(s) => EXIST(a):~a e s]]
http://www.dcproof.com/DrinkersThm1.htm

He doesn't think any further that this is also provable:

/* Also Provable, Right? */
ALL(s):[Set(s) => EXIST(a):[Set(a) & ~a e s]]

And that this isn't provable:

/* Not Provable, Right? */
ALL(s):[Set(s) => EXIST(a):[~Set(a) & ~a e s]]

Here is a new speculation, this is also provable:

ALL(s):[Set(s) => EXIST(a):[Set(a) & a ⊆ s & ~a e s]]

Juding from SEP:

> Despite Quine’s comment, it is possible to see Russell’s paradox in a more positive light.
> the set RB={x∈B:x∉x} exists, but it cannot be an element of B
> https://plato.stanford.edu/entries/russell-paradox/

Mostowski Collapse

unread,
Oct 17, 2022, 3:38:36 AM10/17/22
to
Mostlikely the Set mess started with this paper:

The Quest for Correctness
May 1996 - H. P. Barendregt
https://www.researchgate.net/publication/2293502

First the Cafe is a Type, and then in Fig. 7a there
is suddently U, a non-empty set appears. Whoever draw

Fig. 7a was a moron. Types and Sets are not the same,
further x:Type and x e Set are not the same.

Julio Di Egidio

unread,
Oct 17, 2022, 4:55:23 AM10/17/22
to
Will you insane piece of shit fuck off this thread and
go back play with Dan and co. in your and his threads?!!

You really don't understand the damage you are doing,
here as elsewhere!, do you, you piece of retarded shit,
or you just working for the nazi monster??

ESAD.

Julio

Jeffrey Rubard

unread,
Oct 17, 2022, 4:53:36 PM10/17/22
to
"This is a hoax, right?"

Jeffrey Rubard

unread,
Oct 18, 2022, 3:34:11 PM10/18/22
to
"You don't really think these things, right?"

Jeffrey Rubard

unread,
Oct 19, 2022, 7:05:18 PM10/19/22
to
What does that have to do with the natural numbers, anyhow?

Mostowski Collapse

unread,
Oct 19, 2022, 7:24:37 PM10/19/22
to

65% of italians are communists.

LoL

Ross A. Finlayson

unread,
Oct 19, 2022, 8:27:21 PM10/19/22
to
What's your point?


Barendregt is pretty great, though, the illative is left out as
the same about as the univalent, that instead of having a
real fundamental universe it's "looking either way, the other way".


Barendregt is pretty great though in type theory.

"Janus' introspection" is a notion for a conscientious formalist
how from both his two-faced perspectives, is an inner vision.

Anyways though there's a theory where there's no standard model
of integers, only fragments or extensions.

This re-reads Russell then kicks him out the way.

Mostowski Collapse

unread,
Oct 24, 2022, 6:26:33 AM10/24/22
to
Here have some introspection of the Marxist European south:

The enemy:
transnational security elite

The solution:
The obvious conclusion from the above analysis is that systemic
violence and therefore state repression (if not state terrorism), as
well as counter-violence (including popular terrorism), are built-in
elements of any society characterised by an unequal distribution
of political and economic power. But if the ultimate cause of political
violence in all its forms is the asymmetry of power it is obvious that
the only way out of the cycle of violence is the elimination of this
asymmetry and particularly the elimination of systemic violence.
https://www.inclusivedemocracy.org/dn/vol8/takis_globalwar.htm

Ha Ha, reminds me of Orwells, Animal Farm. What is the impact

of the Ukraine Russian conflict on the Marxist European south?

Mostowski Collapse

unread,
Oct 24, 2022, 6:29:37 AM10/24/22
to
Interesting concept:

Market abolitionism is the belief that the economic market
should be completely eliminated from society. Market abolitionists
argue that markets are ethically abhorrent, antisocial and fundamentally
incompatible with long term human and environmental survival.
https://en.wikipedia.org/wiki/Market_abolitionism

Jeffrey Rubard

unread,
Oct 24, 2022, 3:43:56 PM10/24/22
to
"You know, like that 'journal' n + 1?"

Julio Di Egidio

unread,
Oct 25, 2022, 9:03:09 PM10/25/22
to
You already done, you pieces of retarded polluting shit?!

Keep going you stupid fucks, choke on it...

*Plonk*

Julio

Jeffrey Rubard

unread,
Oct 25, 2022, 10:11:17 PM10/25/22
to
"That's what 'plonk' means."
"I forgot to learn that, exactly. But I could imagine it was so."

Julio Di Egidio

unread,
Oct 25, 2022, 10:21:02 PM10/25/22
to
Just one you can count on, you piece of polluting shit.
Indeed, by all means, keep going, choke on it...

*Plonk*

Julio

Jeffrey Rubard

unread,
Oct 27, 2022, 5:55:26 PM10/27/22
to
"Just one you can count on."
That's not how hoaxes work.
0 new messages