The set of natural numbers is not recursive

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Julio Di Egidio

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Jun 20, 2021, 10:19:11 AM6/20/21
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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

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Jun 20, 2021, 10:58:03 AM6/20/21
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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

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Jun 20, 2021, 11:51:17 AM6/20/21
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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

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Jun 20, 2021, 12:27:22 PM6/20/21
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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

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

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Jun 20, 2021, 5:40:07 PM6/20/21
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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

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Jun 20, 2021, 6:31:00 PM6/20/21
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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

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Jun 20, 2021, 9:06:28 PM6/20/21
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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

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Jun 20, 2021, 9:27:14 PM6/20/21
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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

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Jun 20, 2021, 10:16:57 PM6/20/21
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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

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Jun 21, 2021, 1:46:04 AM6/21/21
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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

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Jun 21, 2021, 3:21:27 AM6/21/21
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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

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Jun 21, 2021, 3:43:26 AM6/21/21
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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

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Jun 21, 2021, 10:06:33 AM6/21/21
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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

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Jun 21, 2021, 12:11:42 PM6/21/21
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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

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Sep 30, 2022, 12:26:02 PMSep 30
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"Starts like empty but is full, ...".

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

Jeffrey Rubard

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Sep 30, 2022, 5:32:28 PMSep 30
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It's... iterative?

Jeffrey Rubard

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Oct 1, 2022, 6:41:11 PMOct 1
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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

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Oct 2, 2022, 7:33:38 PMOct 2
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Sure, I got nothing for you.

Jeffrey Rubard

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

Ross A. Finlayson

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Oct 2, 2022, 10:49:32 PMOct 2
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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

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Oct 2, 2022, 10:52:52 PMOct 2
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Otherwise the rule would be that it is empty, ....

Then there's "half-full".

Ross A. Finlayson

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Oct 2, 2022, 10:59:45 PMOct 2
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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

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Oct 3, 2022, 9:54:50 PMOct 3
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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

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Oct 4, 2022, 12:11:47 AMOct 4
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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

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Oct 4, 2022, 12:27:05 AMOct 4
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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

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Oct 4, 2022, 12:52:21 AMOct 4
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Actually I read a fundamental set theory, number theory, ....

It reads right off, ....

Ross A. Finlayson

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

Jeffrey Rubard

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Oct 6, 2022, 6:42:47 PMOct 6
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That's... not how that works. "Mathematics is mathematics."

Mostowski Collapse

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Oct 7, 2022, 8:00:14 PMOct 7
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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

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Oct 8, 2022, 8:20:32 AMOct 8
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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

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Oct 8, 2022, 10:38:58 AMOct 8
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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

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Oct 8, 2022, 1:54:48 PMOct 8
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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

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Oct 8, 2022, 2:37:15 PMOct 8
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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

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Oct 8, 2022, 3:40:56 PMOct 8
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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

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Oct 8, 2022, 5:32:36 PMOct 8
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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

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Oct 8, 2022, 5:49:25 PMOct 8
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

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Oct 8, 2022, 5:57:05 PMOct 8
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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

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Oct 8, 2022, 7:19:40 PMOct 8
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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

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Oct 9, 2022, 7:22:33 AMOct 9
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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

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Oct 9, 2022, 7:33:49 AMOct 9
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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

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Oct 9, 2022, 7:44:37 AMOct 9
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

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Oct 9, 2022, 7:44:40 AMOct 9
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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

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

Ross A. Finlayson

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Oct 9, 2022, 11:55:56 AMOct 9
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That's what you get for getting led along.

Ross A. Finlayson

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Oct 9, 2022, 12:05:58 PMOct 9
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

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Oct 10, 2022, 5:30:45 AMOct 10
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

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Oct 10, 2022, 5:49:32 AMOct 10