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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

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

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

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

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Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com

Visit my Math Blog at http://www.dcproof.wordpress.com

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:

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:

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:

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

> a recursive set.

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...

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

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

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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.

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
> From Wiki:

> "A is a computable set if and only if A and

> the complement of A are both c.e.."

(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.

--
>>> not a recursive set.

Ben.

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

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quite a very strong one actually. But nobody ever succeeded in

determining even what he was talking about.

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
> 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.

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

SIG MATERIAL! whatever it means

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.

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)"

neither U \ N non empty nor U \ N empty follows.

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

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>>> 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:
> <cut>

>>>> It's easy to see that, as a subset of the universe of numbers, N is

>> "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
>> 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.

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.

Julio

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>
> On Sunday, June 20, 2021 at 2:40:07 PM UTC-7, ju...@diegidio.name wrote:

> > 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?

>

> > 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,
>

> Haltingly halt in looping for example is for static analysis.

it *fails* decidability. Which is pretty much my point.

Julio

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
> 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 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.

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

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

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Graham Cooper haluzinated:

- 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.

"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.

Sep 30, 2022, 12:26:02 PMSep 30

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

Sep 30, 2022, 5:32:28 PMSep 30

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

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"Tell me about it."

Maybe you could read an introductory text on number theory or set theory instead?

Oct 2, 2022, 7:33:38 PMOct 2

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Oct 2, 2022, 7:58:43 PMOct 2

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

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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, ....

Oct 2, 2022, 10:52:52 PMOct 2

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Then there's "half-full".

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

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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, ....

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

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> Yeah, knowing nothing, though, it's been many yeats since I read

> Was sind and was zollen die Zahlen, probably a translation.

>

> Many yeats....

"Yes."

I think you might be a bit of a Philistine, actually.

>

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

>

going "on and on" when you don't even have the basics, like you seem to do

quite a bit.

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

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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.

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

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"almost everywhere", "almost nowhere", "half".

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

Oct 4, 2022, 12:52:21 AMOct 4

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It reads right off, ....

Oct 4, 2022, 1:35:54 AMOct 4

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Oct 6, 2022, 6:42:47 PMOct 6

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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
> Let U be the universe of all numbers.

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?

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

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
> 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.

"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

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?

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

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

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 [...]
> ju...@diegidio.name schrieb am Sonntag, 20. Juni 2021 um 16:19:11 UTC+2:

> >

> > Let U be the universe of all numbers.

> >

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.

> 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..,

> /* Previous Result http://www.dcproof.com/UniversalSet.htm */

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

> Axiom

> 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?

> Whats is the classical way, for example in FOL, to not

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

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

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>
> 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:

> In addition with the axiom

>

> An(n e nat)

>

>

> 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?

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; =]."

spamming idiot...

*Plonk*

Julio

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

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?!
>

> Indeed, PA is, to begin with, a theory of sets, and

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

to

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

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

to

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.)

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]

ALL(s):[Set(s) => EXIST(a):~a ε s]

Set(U)

ALL(a):[a e U]

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

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

1 ALL(s):[Set(s) => EXIST(a):~a ε s]

Axiom

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

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

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)".

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)".

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

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??
> Even DC Spoild itself can derive the contradiction:

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

> Axiom

in the universe of discourse)."

> 2 Set(u)

> Axiom

>

> 3 ALL(a):a ε u

> Axiom

And then I fail to see what is interesting there.

Julio

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

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.

8 ALL(a):a ε u & ~ALL(a):a ε u

Join, 3, 7

is usually considered a contradiction.

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
> 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.

~(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

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

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

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

Oct 9, 2022, 11:52:22 AMOct 9

to

Oct 9, 2022, 11:55:56 AMOct 9

to

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

to

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.

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.
> 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.

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

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