Peano Structures

[Dan Christensen]

Further to discussions here about various supposed models of Peano's Axioms...

Let us define the Peano predicate: Peano(n,s,n0) iff
1. n0 in n
2. s: n --> n
3. s is injective
4. For all x in n: x(x)=/=n0
5. For all subsets P of n: [n0 in P & for all x in P: s(x) in P => P = n]

Suppose Peano(N,S,0) is true, i.e. (N,S,0) in some sense "satisfies" Peano's Axioms.

We have: N = {0, 1, 2, ....} and S(0)=1, S(1)=2, ...

Let E = {0, 2, 4, ...} (the even numbers)

Define f: N --> N such that f(x) = S(S(x)), e.g. f(0)=2, f(1)=3, f(2)=4, ...

Then it would be easy to show that Peano(E,f,0) is also true, i.e. (E,f,0) also satisfies the Peano Axioms.

Clearly, however, E=/=N and f(0)=/=S(0).

Comments?


Dan

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

[David C. Ullrich]

On Thu, 07 Jul 2016 21:44:59 -0700, Dan Christensen wrote:

> Further to discussions here about various supposed models of Peano's
> Axioms...
>
> Let us define the Peano predicate: Peano(n,s,n0) iff
> 1. n0 in n 2. s: n --> n 3. s is injective 4. For all x in n:
> x(x)=/=n0 5. For all subsets P of n: [n0 in P & for all x in P: s(x)
> in P => P = n]
>
> Suppose Peano(N,S,0) is true, i.e. (N,S,0) in some sense "satisfies"
> Peano's Axioms.
>
> We have: N = {0, 1, 2, ....}

If you mean what I think you mean by {0, 1, 2, ....}
then no, this doesn't follow. But never mind that, let's
just assume that N = {0, 1, 2, ....}.

> and S(0)=1, S(1)=2, ...
>
> Let E = {0, 2, 4, ...} (the even numbers)
>
> Define f: N --> N such that f(x) = S(S(x)), e.g. f(0)=2, f(1)=3, f(2)=4,
> ...
>
> Then it would be easy to show that Peano(E,f,0) is also true, i.e.
> (E,f,0) also satisfies the Peano Axioms.

Yes.

>
> Clearly, however, E=/=N and f(0)=/=S(0).
>
> Comments?

Comments on what? You seem to be under the impression that
there's some sort of problem or contradiction or anomaly here.
There is none.

You might clarify things by explaining exactly what problem
you see...

[Dan Christensen]

On Friday, July 8, 2016 at 9:07:33 AM UTC-4, David C. Ullrich wrote:
> On Thu, 07 Jul 2016 21:44:59 -0700, Dan Christensen wrote:
>
> > Further to discussions here about various supposed models of Peano's
> > Axioms...
> >
> > Let us define the Peano predicate: Peano(n,s,n0) iff
> > 1. n0 in n 2. s: n --> n 3. s is injective 4. For all x in n:
> > x(x)=/=n0 5. For all subsets P of n: [n0 in P & for all x in P: s(x)
> > in P => P = n]
> >
> > Suppose Peano(N,S,0) is true, i.e. (N,S,0) in some sense "satisfies"
> > Peano's Axioms.
> >
> > We have: N = {0, 1, 2, ....}
>
> If you mean what I think you mean by {0, 1, 2, ....}
> then no, this doesn't follow. But never mind that, let's
> just assume that N = {0, 1, 2, ....}.
>
> > and S(0)=1, S(1)=2, ...
> >
> > Let E = {0, 2, 4, ...} (the even numbers)
> >
> > Define f: N --> N such that f(x) = S(S(x)), e.g. f(0)=2, f(1)=3, f(2)=4,
> > ...
> >
> > Then it would be easy to show that Peano(E,f,0) is also true, i.e.
> > (E,f,0) also satisfies the Peano Axioms.
>
> Yes.
>
> >
> > Clearly, however, E=/=N and f(0)=/=S(0).
> >
> > Comments?
>
> Comments on what?

Believe or not, there seems to be a school of thought here that E would somehow be identical to N, and f(0)=S(0). Yes, E and N are order isomorphic, but not identical. See the last few postings in the mega-thread, "JG's failed attempt to prove 2+2=4." I really hope I have misunderstood them, because they are sounding as loony as WM and JG here.

Dan

[Me]

On Friday, July 8, 2016 at 6:45:09 AM UTC+2, Dan Christensen wrote:
> Further to discussions here about various supposed models of Peano's Axioms...
>
> Let us define the Peano predicate: Peano(n,s,n0) iff
> 1. n0 in n
> 2. s: n --> n
> 3. s is injective
> 4. For all x in n: x(x)=/=n0
> 5. For all subsets P of n: [n0 in P & for all x in P: s(x) in P => P = n]
>
> Suppose Peano(N,S,0) is true, i.e. (N,S,0) in some sense "satisfies" Peano's Axioms.
>
> We have: N = {0, 1, 2, ....} and S(0)=1, S(1)=2, ...
>
> Let E = {0, 2, 4, ...} (the even numbers)
>
> Define f: N --> N such that f(x) = S(S(x)), e.g. f(0)=2, f(1)=3, f(2)=4, ...
>
> Then it would be easy to show that Peano(E,f,0) is also true, i.e. (E,f,0) also satisfies the Peano Axioms.
>
> Clearly, however, E=/=N and f(0)=/=S(0).
>
> Comments?

Nothing wrong with your approach. In the contex of set theory, Zermelo proposed the set N = {{}, {{}}, {{{}}}, ...} with S(x) = {x} and 0 = {} as a "Model" for the natural numbers. Later von Neumann proposed N = {{}, {{}}, {{} , {{}}}, ...} with S(x) = x u {x} and 0 = {}. (This is the "standard" approach now.)

Actually, one might doubt that there IS such a thing as the "real" natural numbers. This would be an argument against your claim that the Peano axioms _define_ THE natural numbers (after all, you can't select the REAL ones with the help of the Peano axioms). So the Peano axioms "fix" the natural numbers _up to ismomorphism_. And that's indeed good enough for deriving Arithmetics from them (as you've already pointed out several times).

[Jim Burns]

On 7/8/2016 9:30 AM, Dan Christensen wrote:
> On Friday, July 8, 2016 at 9:07:33 AM UTC-4, David C. Ullrich wrote:
>> On Thu, 07 Jul 2016 21:44:59 -0700, Dan Christensen wrote:

>>> Further to discussions here about various supposed models
>>> of Peano's Axioms...
>>>
>>> Let us define the Peano predicate: Peano(n,s,n0) iff
>>> 1. n0 in n 2. s: n --> n 3. s is injective 4.
>>> For all x in n: x(x)=/=n0 5. For all subsets P of n:
>>> [n0 in P & for all x in P: s(x) in P => P = n]
>>>
>>>
>>> Suppose Peano(N,S,0) is true, i.e. (N,S,0) in some sense
>>> "satisfies" Peano's Axioms.

"Satisfies" not in just any sense, but in this sense:

<quote SEP>[ASCIIfied]
Sometimes we write or speak a sentence S that expresses nothing
either true or false, because some crucial information is missing
what the words mean. If we go on to add this information, so that
S comes to express a true or false statement, we are said to
_interpret_ S , and the added information is called an
_interpretation_ of S . If the interpretation I happens to make
S state something true, we say that I is a _model_ of S , or
that I _satisfies_ S , in symbols 'I |= S'. Another way of saying
that I is a model of S is to say that S is _true in_ I , and
so we have the notion of _model-theoretic truth+ , which is truth
in a particular interpretation. But one should remember that the
statement ' S is true in I ' is just a paraphrase of ' S , when
interpreted as in I , is true '; so model-theoretic truth is
parasitic on plain ordinary truth, and we can always paraphrase
it away.
</quote SEP>
http://plato.stanford.edu/entries/model-theory/#Basic

If you're using scare quotes for "satisfies" you either don't
know or refuse to admit you know what you're talking about.

If you refuse to fix that, that is pretty much a defining
characteristic of an internet kook: someone who insists on
talking as though they know what they're talking about
in the face of overwhelming evidence that they don't.
(Note: this is not the same as being ignorant or wrong,
things that happens to every mortal.)

>>> We have: N = {0, 1, 2, ....}
>>
>> If you mean what I think you mean by {0, 1, 2, ....}
>> then no, this doesn't follow. But never mind that, let's
>> just assume that N = {0, 1, 2, ....}.
>>
>>> and S(0)=1, S(1)=2, ...
>>>
>>> Let E = {0, 2, 4, ...} (the even numbers)
>>>
>>> Define f: N --> N such that f(x) = S(S(x)),
>>> e.g. f(0)=2, f(1)=3, f(2)=4, ...
>>>
>>> Then it would be easy to show that Peano(E,f,0) is also true,
>>> i.e. (E,f,0) also satisfies the Peano Axioms.
>>
>> Yes.
>>
>>>
>>> Clearly, however, E=/=N and f(0)=/=S(0).
>>>
>>> Comments?
>>
>> Comments on what?
>
> Believe or not, there seems to be a school of thought here that
> E would somehow be identical to N, and f(0)=S(0).

Do you have any examples of anyone else saying this, Dan?
All I have seen are people trying to correct you when you
say this. And I certainly have not said this.

This could be an honest mistake on your part, though an
honest mistake repeated over and over the the face of multiple
corrections.

Or it could be a less-than-honest mistake. In which case you
care nothing about finding out what is true and are only
interested in "winning" in some sense. This too is a defining
characteristic of the internet kook.

> Yes, E and N
> are order isomorphic, but not identical. See the last few postings
> in the mega-thread, "JG's failed attempt to prove 2+2=4."
> I really hope I have misunderstood them, because they are sounding
> as loony as WM and JG here.

Be at peace, Dan. You have misunderstood us.

[Dan Christensen]

> Actually, one might doubt that there IS such a thing as the "real" natural numbers. This would be an argument against your claim that the Peano axioms _define_ THE natural numbers (after all, you can't select the REAL ones with the help of the Peano axioms). So the Peano axioms "fix" the natural numbers _up to ismomorphism_. And that's indeed good enough for deriving Arithmetics from them (as you've already pointed out times).

I have noticed that simply stating Peano's Axioms at the beginning of a formal proof pretty much "fixes" the natural numbers, period. You can use existential instantiation (specification) to attach any number of labels to S(1), for example, and they would all be equal to S(1) and to one another. But you will not be able to re-use them to label other expressions -- a restriction imposed the the EI rule. You could also not construct another set and call it N, or another function and call it S.

[Peter Percival]

Dan thinks that Peano's axioms (second order) define the natural numbers
and when a number of isomorphic but not identical structures are shown
to him that that satisy the axioms he gets in a muddle. E.g., these
structures have been discussed

{1, 2, 3, ...} with successor x|->x+1

and

{1, 1/2, 1/4, ...} with successor x|->x/2 .

Those who claim that both of them satisfy Peano's axioms are accused of
claiming that 2 = 1/2.

This has been going on in the thread "JG's failed attempt to prove
2+2=4" for weeks.



--
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.
Ulysses, Alfred, Lord Tennyson

[Peter Percival]

Dan Christensen wrote:

> I have noticed that simply stating Peano's Axioms at the beginning of
> a formal proof pretty much "fixes" the natural numbers, period.

You've noticed that have you? Have you not also noticed that there are
many different structures that satisfy Peano's axioms?

[Peter Percival]

Me wrote:

> Nothing wrong with your approach. In the contex of set theory,
> Zermelo proposed the set N = {{}, {{}}, {{{}}}, ...} with S(x) = {x}
> and 0 = {} as a "Model" for the natural numbers. Later von Neumann
> proposed N = {{}, {{}}, {{} , {{}}}, ...} with S(x) = x u {x} and 0 =
> {}. (This is the "standard" approach now.)
>
> Actually, one might doubt that there IS such a thing as the "real"
> natural numbers. This would be an argument against your claim that
> the Peano axioms _define_ THE natural numbers (after all, you can't
> select the REAL ones with the help of the Peano axioms). So the Peano
> axioms "fix" the natural numbers _up to ismomorphism_. And that's
> indeed good enough for deriving Arithmetics from them (as you've
> already pointed out several times).

One might draw Dan's attention to Benacerraf's /What numbers could not be/.

[Dan Christensen]

On Friday, July 8, 2016 at 10:33:28 AM UTC-4, Jim Burns wrote:

> >
> > Believe or not, there seems to be a school of thought here that
> > E would somehow be identical to N, and f(0)=S(0).
>
> Do you have any examples of anyone else saying this, Dan?
> All I have seen are people trying to correct you when you
> say this. And I certainly have not said this.
>

So, that was an impostor who said that {1, 1/2, 1/4, ...} and the set of natural numbers are really "the same thing." Whew! For a while there, I thought you were really losing it, Jim.

Dan

[Dan Christensen]

On Friday, July 8, 2016 at 11:02:06 AM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
> > I have noticed that simply stating Peano's Axioms at the beginning of
> > a formal proof pretty much "fixes" the natural numbers, period.
>
> You've noticed that have you? Have you not also noticed that there are
> many different structures that satisfy Peano's axioms?
>

Irrelevant in this context. Given only these properties of the natural numbers (that can be seen as essentially defining them), can derive all of modern number theory. You don't want to call it a "definition?" How about simply, the essential properties of the natural numbers? Too definition-like for you? Oh, well.

Dan

[Dan Christensen]

Several times, it was claimed (perhaps by an impostor?) that {1, 1/2, 1/4, ... } were the "same thing" as the set of natural numbers. JG couldn't have said it better himself. Hmmm...


Dan

[Peter Percival]

And several times it was claimed (perhaps by a mental defective?) that
those who point that both of the above structures satisfy Peano's axioms
were maintaining that 2 = 1/2.

> ) that {1, 1/2, 1/4, ... } were the "same thing" as the set of natural numbers. JG couldn't have said it better himself. Hmmm...
>
>
> Dan
>

[Peter Percival]

Dan Christensen wrote:
> On Friday, July 8, 2016 at 11:02:06 AM UTC-4, Peter Percival wrote:
>> Dan Christensen wrote:
>>
>>> I have noticed that simply stating Peano's Axioms at the
>>> beginning of a formal proof pretty much "fixes" the natural
>>> numbers, period.
>>
>> You've noticed that have you? Have you not also noticed that there
>> are many different structures that satisfy Peano's axioms?
>>
>
>
> Irrelevant in this context. Given only these properties of the
> natural numbers (that can be seen as essentially defining them),

You're seeing too much. The name Dan doesn't define you because there
is more than one person called Dan.

> can
> derive all of modern number theory. You don't want to call it a
> "definition?"

Would you want to call "Dan" a definition of you?

> How about simply, the essential properties of the
> natural numbers? Too definition-like for you? Oh, well.
>
> Dan
>

[Jussi Piitulainen]

Peter Percival writes:

> Me wrote:
>
>> Nothing wrong with your approach. In the contex of set theory,
>> Zermelo proposed the set N = {{}, {{}}, {{{}}}, ...} with S(x) = {x}
>> and 0 = {} as a "Model" for the natural numbers. Later von Neumann
>> proposed N = {{}, {{}}, {{} , {{}}}, ...} with S(x) = x u {x} and 0 =
>> {}. (This is the "standard" approach now.)
>>
>> Actually, one might doubt that there IS such a thing as the "real"
>> natural numbers. This would be an argument against your claim that
>> the Peano axioms _define_ THE natural numbers (after all, you can't
>> select the REAL ones with the help of the Peano axioms). So the Peano
>> axioms "fix" the natural numbers _up to ismomorphism_. And that's
>> indeed good enough for deriving Arithmetics from them (as you've
>> already pointed out several times).
>
> One might draw Dan's attention to Benacerraf's /What numbers could not
> be/.

Colin Mclarty, /Numbers can be just what they have to/, in response.

[abu.ku...@gmail.com]

Thanks, coliN\

[Jussi Piitulainen]

Hm. I don't know why Gnus cannot quite decide whether to show the
slashes around the title or not - it tends to be too clever for me - and
I mispelt McLarty's name anyway. Sorry about the mispeling. I'll try and
capitalize even the title of the paper as faithfully as I can now.

Colin McLarty, "Numbers Can Be Just What They Have To", Noûs 27:4 (1993)
487-498. The author was at Case Western Reserve University at the time,
there's a PDF there now, and I hope I won't need to regret spelling the
name of the journal in a way that seems to me to be correct, with the u
with the hat :)

http://www.cwru.edu/artsci/phil/NumbersCanBeJustWhattheyHaveTo.pdf

[Me]

On Friday, July 8, 2016 at 5:06:57 PM UTC+2, Peter Percival wrote:

> One might draw Dan's attention to Benacerraf's /What numbers could not be/.

I've read that paper (some time ago). Wasn't that satisfied with it (to say the least). Actually, I got rather angry while reading it. Well...

[WM]

Am Freitag, 8. Juli 2016 16:59:59 UTC+2 schrieb Peter Percival:



>
> Dan thinks that Peano's axioms (second order) define the natural numbers
> and when a number of isomorphic but not identical structures are shown
> to him that that satisy the axioms he gets in a muddle. E.g., these
> structures have been discussed
>
> {1, 2, 3, ...} with successor x|->x+1
>
> and
>
> {1, 1/2, 1/4, ...} with successor x|->x/2 .

and

1, pi, pipi, pipipi, ...

and

1, Dan, wormhole, 2, 3, 4, ...

and

a sequence of nxn-matrices

and

a sequence of n-th order equations

and

infinitely many sequences which have no numbers as terms.

>
> Those who claim that both of them satisfy Peano's axioms are accused of
> claiming that 2 = 1/2.

Regards, WM

[Peter Percival]

those who point out that

sorry

[Virgil]

In article <2ffac743-0262-4673...@googlegroups.com>,

Actually the Peano axioms define any and all ordered sets which are
order-isomorphic to the naturally ordered naturals.

Collectively they are known as inductive sets.
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

[Me]

On Saturday, July 9, 2016 at 12:23:12 AM UTC+2, Virgil wrote:

> Collectively they are known as inductive sets.

Jep.

[Jim Burns]

The school of thought you are thinking of has told you
that E and N are _isomorphic_ not _identical_ . And told you.
And told you.

It is _you_ , Dan, who declare that the Peano axioms are the
definition of the natural numbers and _at the same time_
something which satisfies the Peano axioms _is not_ the
natural numbers, in particular:
( { 1, 1/3, 1/4, ... }, 1, x |-> x/2 )

For a number of posts now, I haven't tried to tell you
_anything_ is the natural numbers, I've only been repeating
back to you what _you've_ said is the natural numbers.
You don't like what you've said? That's not my problem.

It's true that earlier in our exchange, I tried to say
"This is the natural numbers" in a way compatible with
what you're saying, but not for a while now. You've
got too much to un-learn before there'd be any point to
that conversation.
For example, "Interpretation I satisfies theory T "
is not informal.

[Dan Christensen]

On Friday, July 8, 2016 at 12:24:25 PM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
> > On Friday, July 8, 2016 at 11:02:06 AM UTC-4, Peter Percival wrote:
> >> Dan Christensen wrote:
> >>
> >>> I have noticed that simply stating Peano's Axioms at the
> >>> beginning of a formal proof pretty much "fixes" the natural
> >>> numbers, period.
> >>
> >> You've noticed that have you? Have you not also noticed that there
> >> are many different structures that satisfy Peano's axioms?
> >>
> >
> >
> > Irrelevant in this context. Given only these properties of the
> > natural numbers (that can be seen as essentially defining them),
>
> You're seeing too much. The name Dan doesn't define you because there
> is more than one person called Dan.
>
> > can
> > derive all of modern number theory. You don't want to call it a
> > "definition?"
>
> Would you want to call "Dan" a definition of you?
>
> > How about simply, the essential properties of the
> > natural numbers? Too definition-like for you? Oh, well.
> >

Do you agree that Peano's Axioms for (N,S,0) gives us the essential properties of the set of natural numbers to the extent that all of modern number theory can be derived from them using nothing more than the ordinary rules of logic and set theory? And that any theorem derived for a structure (N',S',0') that is order-isomorphic to (N,S,0) can be translated to a theorem in (N,S,0) simply be substituting N for N', S for S' and 0 for 0'?

[Peter Percival]

I'm not sure what "all of modern number theory" is. Naturally, I am
worried about incompleteness. Also, as remarked before, with set
theory, Peano's axioms, qua axioms, are redundant: they are theorems of
set theory. I recall you expressing doubts about the axiom of infinity,
and when I suggested replacing it with its negation, this exchange -

Me: ZF-Inf+~Inf is equivalent to PA.

You: Nonsense.

Me: Really? Why is the claim that ZF-Inf+~Inf is equivalent to PA nonsense?

You: You will not be able to derive Peano's Axioms from it.

Me: Let phi be a formula in the language of ZF, then there is a
translation, phi*, of phi into the language of PA such that

ZF-Inf+~Inf |- phi iff PA |- phi*.

You: Gibberish!

Me: Let psi be a formula in the language of PA, then there is a
translation, psi', of psi into the language of ZF (it's the inverse of
*) such that

ZF-Inf+~Inf |- psi' iff PA |- psi.

You: Ditto.

resulted. (See 8/6/2016, 19;43, in the thread "The arguments against
set theory are invalid as soon as ...".) Why nonsense, and why gibberish?

> And that any theorem
> derived for a structure (N',S',0') that is order-isomorphic to
> (N,S,0)

Why *order*-isomophic? (N',S',0') and (N,S,0) being isomorphic means
there is a one-to-one onto map f:N' -> N such that
i) S(fx) = fS'(x) for all x in N', and
ii) f0' = 0.

> can be translated to a theorem in (N,S,0) simply be
> substituting N for N', S for S' and 0 for 0'?
>
>
> Dan
>
> Download my DC Proof 2.0 software at http://www.dcproof.com Visit my
> Math Blog at http://www.dcproof.wordpress.com
>

[WM]

Am Samstag, 9. Juli 2016 14:20:12 UTC+2 schrieb Dan Christensen:

>
> Do you agree that Peano's Axioms for (N,S,0) gives us the essential properties of the set of natural numbers to the extent that all of modern number theory can be derived from them using nothing more than the ordinary rules of logic and set theory?

No, the Peano axioms supply the set of words of the Bible and their repetitions. These have no numerical value and have nothing to do with mathematics. They supply nothing of number theory. Nevertheless the rules of set theory and logic do not deny or exclude that model.

Regards, WM

[Dan Christensen]

On Saturday, July 9, 2016 at 9:00:00 AM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:

> >
> > Do you agree that Peano's Axioms for (N,S,0) gives us the essential
> > properties of the set of natural numbers to the extent that all of
> > modern number theory can be derived from them using nothing more than
> > the ordinary rules of logic and set theory?
>
> I'm not sure what "all of modern number theory" is. Naturally, I am
> worried about incompleteness.

I doubt that number theorists have even given it a second thought.

> Also, as remarked before, with set
> theory, Peano's axioms, qua axioms, are redundant: they are theorems of
> set theory. I recall you expressing doubts about the axiom of infinity,

I don't have one in my own system. I find Peano's Axioms much more straightforward and intuitive than an axiom of infinity.


> and when I suggested replacing it with its negation...

Yeah, that was weird.

[snip]

>
> > And that any theorem
> > derived for a structure (N',S',0') that is order-isomorphic to
> > (N,S,0)
>
> Why *order*-isomophic? (N',S',0') and (N,S,0) being isomorphic means
> there is a one-to-one onto map f:N' -> N such that
> i) S(fx) = fS'(x) for all x in N', and
> ii) f0' = 0.
>

Yes, if Peano(N,S,0) and Peano(N',S',0') (see OP), then the structures (N,S,0) and (N',S',0') are order-isopmorhpic, i.e. there exists a bijection f: N <--> N' such that:

0 <--> 0'
S(0) <--> S'(0')
S(S(0)) <--> S'(S'(0'))
...

[Peter Percival]

Dan Christensen wrote:
> On Saturday, July 9, 2016 at 9:00:00 AM UTC-4, Peter Percival wrote:
>> Dan Christensen wrote:
>
>>>
>>> Do you agree that Peano's Axioms for (N,S,0) gives us the essential
>>> properties of the set of natural numbers to the extent that all of
>>> modern number theory can be derived from them using nothing more than
>>> the ordinary rules of logic and set theory?
>>
>> I'm not sure what "all of modern number theory" is. Naturally, I am
>> worried about incompleteness.
>
> I doubt that number theorists have even given it a second thought.
>
>
>> Also, as remarked before, with set
>> theory, Peano's axioms, qua axioms, are redundant: they are theorems of
>> set theory. I recall you expressing doubts about the axiom of infinity,
>
> I don't have one in my own system. I find Peano's Axioms much more straightforward and intuitive than an axiom of infinity.
>
>
>> and when I suggested replacing it with its negation...
>
> Yeah, that was weird.

Oh, just weird? Not nonsense and gibberish?

> [snip]

Is it your belief, you vile mother fucking syphilitic cunt, that if you
snip something it no longer exists?

>>
>>> And that any theorem
>>> derived for a structure (N',S',0') that is order-isomorphic to
>>> (N,S,0)
>>
>> Why *order*-isomophic? (N',S',0') and (N,S,0) being isomorphic means
>> there is a one-to-one onto map f:N' -> N such that
>> i) S(fx) = fS'(x) for all x in N', and
>> ii) f0' = 0.
>>
>
> Yes, if Peano(N,S,0) and Peano(N',S',0') (see OP), then the structures (N,S,0) and (N',S',0') are order-isopmorhpic, i.e. there exists a bijection f: N <--> N' such that:
>
> 0 <--> 0'
> S(0) <--> S'(0')
> S(S(0)) <--> S'(S'(0'))
> ...

Ok, so I'll tell you what order isomorphism is. (X, <) and (X', <'),
are order isomorphic if there is a one-to-one onto map f:X -> X' such
that x < y iff fx <' fy for all x,y in X. But don't let that bother
you; you just carry on using "order-isomorphic" in your way.

[Jim Burns]

On 7/9/2016 8:20 AM, Dan Christensen wrote:
[to Peter Percival ]

>> Dan Christensen wrote:

>>> How about simply, the essential properties of the
>>> natural numbers? Too definition-like for you? Oh, well.
>
> Do you agree that Peano's Axioms for (N,S,0) gives us
> the essential properties of the set of natural numbers
> to the extent that all of modern number theory can be derived
> from them using nothing more than the ordinary rules of
> logic and set theory?

The Peano axioms give us what they give us. If those are
the properties that you want the natural numbers to have,
then I'd say it's at least not terrible.
(There was this poster once-upon-a-time who defined
the natural numbers as pretty much any infinite set,
and the _familiar_ +, *, < as pretty much anything.
That, I would call terrible.)

If you want to argue from the essential-ness of those properties
that PA _should be_ taken as defining the natural numbers,
then you should say what you mean by "essential" in a
non-circular way -- and then give your argument.

> And that any theorem derived for a structure (N',S',0')
> that is order-isomorphic to (N,S,0) can be translated to
> a theorem in (N,S,0) simply be substituting N for N',
> S for S' and 0 for 0'?

Do you agree that _does not_ mean this?
S(S(0)) = S'(S'(0')) [??]

Rather, it _does_ mean, for the isomorphism g: N -> N'
g( S(S(0)) ) = S'(S'(0'))

Then, for the two structures
( { 1, 2, 3, ... }, 1, x |-> x+1 )
and
( { 1, 1/2, 1/4, ... }, 1, x |-> x/2 )
which both satisfy the Peano axioms, there exists an
isomorphism g, where g(1) = 1 and g(x+1) = g(x)/2

Then we would have g(2) = 1/2 -- which does not mean 2 = 1/2 .

If you want to say that the Peano axioms define the
natural numbers _up to isomorphism_ then I would agree.
What does "define _up to isomorphism_ " mean?

Consider the _numerals_ for the natural numbers.
In base ten, we have 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
In base two, we have 0, 1, 10, 11, 100, 101, 110, 111, ...
They are both perfectly fine ways (among many) to label
natural numbers. _There is no one way_ to do that.
In one context, S(S(0)) = 2 .
In another context, S(S(0)) = 10 .
This does not mean (the numeral) 2 = (the numeral) 10 .

There is an isomorphism between decimal numerals and
binary numerals that preserves everything we care about
as we change from one labeling scheme to another. So,
whether we are using decimal, binary, or some other
kind of numerals to describe natural numbers is something
that just doesn't get much attention. You pick what
numeral system you want, and I'll pick whatever numeral
system I want, and if they're different, we'll translate
back and forth. So, which we use doesn't get attention because
it really doesn't deserve attention.

In the same way, which isomorphic model of PA we are talking
about really doesn't deserve attention (and all models of PA
are isomorphic). They are all equally good, and we can translate
back and forth as needed.

I should point out that, as reasonable as this sounds, it
leads us to calling 1/2 a natural number _up to isomorphism_
because models of PA exist in which 1/2 plays the role of
a natural number. You, Dan, have been maybe a little resistant
to this conclusion.

If you have some other idea as to what it means to be a
natural number, _period_ and not _up to isomorphism_ ,
I haven't seen it from you yet. Ridiculing the idea of
1/2 being a natural number says what you think it is _not_ .
It doesn't say what it _is_ , period.

[Virgil]

In article <4e69753b-6098-4012...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Samstag, 9. Juli 2016 14:20:12 UTC+2 schrieb Dan Christensen:
>
> >
> > Do you agree that Peano's Axioms for (N,S,0) gives us the essential
> > properties of the set of natural numbers to the extent that all of modern
> > number theory can be derived from them using nothing more than the ordinary
> > rules of logic and set theory?
>
> No, the Peano axioms supply the set of words of the Bible and their
> repetitions.

And with the the structure of an infinite well-ordered set with unique
non-successor.

>These have no numerical value

But they all have the order structure both neccessary and sufficient for
building the ordered set of natural numbers

> They supply nothing of number theory.

Except the model for it, an infinite well ordered set with unique
non-successor.

> Nevertheless the rules of
> set theory and logic do not deny or exclude that model.

And no well-ordered set either without a unique non-successor or with a
last element (thus no non-Peano set) can model the naturals.

In Mathematics, Cantor is provably right,
Card(Q) = Card(N) and Card(N) < Card(R)
So Card(Q) < Card(R), with fewer rationals than reals

In WMaths/WMytheology, WM claims Cantor is wrong,
WM claims Card(Q) > Card(N) and Card(N) = Card(R)
So WM claims Card(Q) > Card(R), with more rationals than reals.

But there aren't any rationals that are not reals in mathematics,
only in WM's witless worthless wacky anti-mathemtical world of
WMytheology.

[Me]

On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:

> No, the Peano axioms supply the set of words of the Bible and their
> repetitions.

No, idiot, the don't "supply" anything. Moreover, since the set of words in the Bible is FINITE, it can't be used as a denotation of "IN" in the Peano axioms.

[WM]

Am Samstag, 9. Juli 2016 19:53:52 UTC+2 schrieb Virgil:


> > No, the Peano axioms supply the set of words of the Bible and their
> > repetitions.
>
> And with the the structure of an infinite well-ordered set with unique
> non-successor.

That structure is given by counting +1 in a much better form.

>
> >These have no numerical value
>
> But they all have the order structure both neccessary and sufficient for
> building the ordered set of natural numbers

That structure is given by counting +1 in a much better form.

Regards, WM

[WM]

Am Samstag, 9. Juli 2016 19:24:36 UTC+2 schrieb Dan Christensen:

> On Saturday, July 9, 2016 at 9:14:42 AM UTC-4, WM wrote:
> > Am Samstag, 9. Juli 2016 14:20:12 UTC+2 schrieb Dan Christensen:
> >
> > >
> > > Do you agree that Peano's Axioms for (N,S,0) gives us the essential properties of the set of natural numbers to the extent that all of modern number theory can be derived from them using nothing more than the ordinary rules of logic and set theory?
> >
> > No, the Peano axioms supply the set of words of the Bible and their repetitions. These have no numerical value and have nothing to do with mathematics.
>

> This from an

From whom it comes is irrelevant. The question is whether it is true.

> > They supply nothing of number theory.

> Peano's Axioms, on the other can be used to derive all of modern number theory.

So you think it is not true.
>
> You must be so jealous!

Do you think?

> > Nevertheless the rules of set theory and logic do not deny or exclude that model.
>

> They do if the set theory has the axiom of infinity.

An argument after all? But no. The words of the Bible can be continued (for instance by doubling them) in infinity.

Regards, WM

[Peter Percival]

"and their repetitions".

[Virgil]

In article <c8c8f797-873b-4ff6...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Samstag, 9. Juli 2016 19:53:52 UTC+2 schrieb Virgil:
>
>
> > > No, the Peano axioms supply the set of words of the Bible and their
> > > repetitions.
> >
> > And with the the structure of an infinite well-ordered set with unique
> > non-successor.
>
> That structure is given by counting +1 in a much better form.

Not without being given either 0 or 1 as a starting point and the lack
of an ending point. Both of which Peano provides but WM does not.

> >
> > >These have no numerical value
> >
> > But they all have the order structure both neccessary and sufficient for
> > building the ordered set of natural numbers
>
> That structure is given by counting +1 in a much better form.

Not without being given either 0 or 1 as a starting point and the lack
of an ending point. Both of which Peano provides but WM does not.

In Mathematics, Cantor is provably right,

with Card(Q) = Card(N) and Card(N) < Card(R)

So Card(Q) < Card(R), with fewer rationals than reals

In WMaths/WMytheology, WM claims Cantor is wrong,

WM claims both Card(Q) > Card(N) and Card(N) = Card(R)
requiring Card(Q) > Card(R), with more rationals than reals.
But there are not any rational that re not reals anywhere outside of
WM's witless worthless wacky world of WMytheology.

So WM's witless worthless wacky world of WMytheology is WRONG!

[Virgil]

In article <22856ea7-943b-432f...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:


> From whom it comes is irrelevant. The question is whether it is true.

From WM comes WM's claims Cantor is always wrong,
Thus WM claims Card(Q) > Card(N) and Card(N) = Card(R)
So Card(Q) > Card(R), requiring more rationals than reals.

But in any mathematics free from WM's witless worthless wacky world of
WMytheology, there aren't any rationals that are not real

Thus WM's witless worthless wacky world of WMytheology is full of lies!

[Dan Christensen]

On Saturday, July 9, 2016 at 12:31:32 PM UTC-4, Jim Burns wrote:
> On 7/9/2016 8:20 AM, Dan Christensen wrote:
> [to Peter Percival ]
> >> Dan Christensen wrote:
>
> >>> How about simply, the essential properties of the
> >>> natural numbers? Too definition-like for you? Oh, well.
> >
> > Do you agree that Peano's Axioms for (N,S,0) gives us
> > the essential properties of the set of natural numbers
> > to the extent that all of modern number theory can be derived
> > from them using nothing more than the ordinary rules of
> > logic and set theory?
>
> The Peano axioms give us what they give us. If those are
> the properties that you want the natural numbers to have,
> then I'd say it's at least not terrible.

Evasive.

> (There was this poster once-upon-a-time who defined
> the natural numbers as pretty much any infinite set,
> and the _familiar_ +, *, < as pretty much anything.
> That, I would call terrible.)
>
> If you want to argue from the essential-ness of those properties
> that PA _should be_ taken as defining the natural numbers,
> then you should say what you mean by "essential" in a
> non-circular way -- and then give your argument.
>

Each of Peano's Axioms is essential in the sense that removing or negating it would not allow us to derive modern number theory as we know it. Now please answer the above question.

[snip]

[Dan Christensen]

On Saturday, July 9, 2016 at 5:41:25 PM UTC-4, WM wrote:
> Am Samstag, 9. Juli 2016 19:24:36 UTC+2 schrieb Dan Christensen:
> > On Saturday, July 9, 2016 at 9:14:42 AM UTC-4, WM wrote:
> > > Am Samstag, 9. Juli 2016 14:20:12 UTC+2 schrieb Dan Christensen:
> > >
> > > >
> > > > Do you agree that Peano's Axioms for (N,S,0) gives us the essential properties of the set of natural numbers to the extent that all of modern number theory can be derived from them using nothing more than the ordinary rules of logic and set theory?
> > >
> > > No, the Peano axioms supply the set of words of the Bible and their repetitions. These have no numerical value and have nothing to do with mathematics.
> >

> > This from an idiot who mistakenly took a statement of Peano's induction axiom for the definition of the natural numbers while rejecting the other axioms.

>
> From whom it comes is irrelevant. The question is whether it is true.
>

It isn't true. It is nonsense.

> > > They supply nothing of number theory.
>

WM snipped:

No, it is your goofy number system that supplies nothing, not even the most elementary results of basic arithmetic -- not 2+2=4, 1=/=2, or even the existence a single number! Truly a dead end.

> > Peano's Axioms, on the other can be used to derive all of modern number theory.
>
> So you think it is not true.

No, I think it is true.

> >
> > You must be so jealous!
>
> Do you think?
>

Yes, and it seems to be making you crazy, Mucke.

> > > Nevertheless the rules of set theory and logic do not deny or exclude that model.
> >
> > They do if the set theory has the axiom of infinity.
>
> An argument after all? But no. The words of the Bible can be continued (for instance by doubling them) in infinity.
>

You are losing it here, Mucke. Maybe you need a little vacation away from the internet to gather your thoughts.

[WM]

Am Samstag, 9. Juli 2016 23:38:37 UTC+2 schrieb Me:
> On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:
>
> > No, the Peano axioms supply the set of words of the Bible and their
> > repetitions.
>

> since the set of words in the Bible is FINITE, it can't be used as a denotation of "IN" in the Peano axioms.

I pointed to their infinite repetitions, combinations, or concatenations. They are sufficient. But in order to use them as a denotation of |N, you must first know |N. The Peano axioms don't supply that meaning, in particular they do not prove the constant distance.

REgards, WM

[WM]

Am Sonntag, 10. Juli 2016 06:57:41 UTC+2 schrieb Dan Christensen:


> Each of Peano's Axioms is essential in the sense that removing or negating it would not allow us to derive modern number theory as we know it.

Of course all axioms are required to remedy Peano's clumsy successless successor approach at least to a certain level. But the natural numbers are not defined by his axioms.

Regards, WM

[WM]

Am Sonntag, 10. Juli 2016 07:09:27 UTC+2 schrieb Dan Christensen:


> > > > No, the Peano axioms supply the set of words of the Bible and their repetitions. These have no numerical value and have nothing to do with mathematics.
> > >
> > > This from an idiot who mistakenly took a statement of Peano's induction axiom for the definition of the natural numbers while rejecting the other axioms.
> >
> > From whom it comes is irrelevant. The question is whether it is true.
> >
>
> It isn't true. It is nonsense.

Which axiom does rule it out?

>

> You are losing it here,

No I gain an incredible example.

Regards, WM

[WM]

Am Sonntag, 10. Juli 2016 02:56:57 UTC+2 schrieb Virgil:


> > That structure is given by counting +1 in a much better form.
>
> Not without being given either 0 or 1 as a starting point and the lack
> of an ending point. Both of which Peano provides but WM does not.

Wrong. I give 1 in |N. And every n is not the last one by the existence of n+1.

Regards, WM

[Me]

On Saturday, July 9, 2016 at 11:53:35 PM UTC+2, Peter Percival wrote:
> Me wrote:
> > On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:
> >
> >> No, the Peano axioms supply the set of words of the Bible and their
> >> repetitions.
> >
> > No, idiot, the don't "supply" anything. Moreover, since the set of words in the Bible is FINITE, it can't be used as a denotation of "IN" in the Peano axioms.
>
> "and their repetitions".

Yeah, idiot, I read that. *lol* You are dumb like shit!

A SET OF WORDS contains "repetitions", are you SERIOUS, OR JUST AN ASSHOLE FULL OF SHIT?

[Me]

On Saturday, July 9, 2016 at 11:41:25 PM UTC+2, WM wrote:

> The words of the Bible can be continued (for instance by doubling them)
> in infinity.

Huh? You mentioned THE SET OF WORDS in the Bible. That "it can be continued" was never up for discussion, idiot.

[Me]

On Sunday, July 10, 2016 at 10:58:37 AM UTC+2, WM wrote:
> Am Samstag, 9. Juli 2016 23:38:37 UTC+2 schrieb Me:
> > On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:
> > >
> > > No, the Peano axioms supply the set of words of the Bible and their
> > > repetitions.
> > >
> > since the set of words in the Bible is FINITE, it can't be used as a denotation of "IN" in the Peano axioms.
>
> I pointed to their infinite repetitions, combinations, or concatenations.

No, you didn't. Moroever, A SET can't contain "repetitions" of elements, idiot.

On the other and of we start with a FINITE ALPHABET we certainly can have a INFINITE set of words (from this aplhabet).

We might consider the set B = {"I", "II", "III", IIII", ...} for example. Where the alphabet just consists of "I".

> They are sufficient. But in order to use them as a denotation of |N, you
> must first know |N.

Huh? All we need is a MODEL for the Peano axioms: Just let N := B, S(X) the concatenation of X with "I". and 1 := "I".

> The Peano axioms don't supply that meaning, in particular they do not
> prove the constant distance.

You are talking nonsense, man.

We can PROVE from that Peano axioms that

for every n e IN:
there is no x e IN such that n < x < S(n) ,

after defining < the usual way. With other words, we can PROVE that for any element n in IN there is no element in IN such that this element is "between" n and S(n). Got that?

Moreover we can PROVE that for every n e IN:

S(n) - n = 1 ,

after defining + (and then -) the usual way. With other words, we can PROVE that the "distance" between any element n in IN and its successor S(n) is constant (and equals to 1). Got that?

Maybe you prefer the following form:

An e IN Am e IN: S(n) - n = S(m) - m ,

the "distance" between S(x) and x is "constant".

[Me]

On Sunday, July 10, 2016 at 11:13:17 AM UTC+2, WM wrote:

> I give 1 in |N.

WM you are just a liar or dumb like shit. Either way, NO, you DONT'T "give" that in your "axiom system for the natural numbers".

Otherwise, can you please QUOTE that axiom?

[Me]

On Sunday, July 10, 2016 at 11:09:29 AM UTC+2, WM wrote:
> Am Sonntag, 10. Juli 2016 07:09:27 UTC+2 schrieb Dan Christensen:
> > > >
> > > > This from an idiot who mistakenly took a statement of Peano's
> > > > induction axiom for the definition of the natural numbers while
> > > > rejecting the other axioms.

Right. That's indded the case.

> > > The question is whether it is true.

Yes, it is true that you mistakenly took a statement of Peano's induction axiom for the definition of the natural numbers.

> Which axiom does rule it out?

Huh?

From your "axiom system" we cannot prove 1 e IN, for example.

[Peter Percival]

Me wrote:
> On Saturday, July 9, 2016 at 11:53:35 PM UTC+2, Peter Percival wrote:
>> Me wrote:
>>> On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:
>>>
>>>> No, the Peano axioms supply the set of words of the Bible and their
>>>> repetitions.
>>>
>>> No, idiot, the don't "supply" anything. Moreover, since the set of words in the Bible is FINITE, it can't be used as a denotation of "IN" in the Peano axioms.
>>
>> "and their repetitions".
>
> Yeah, idiot, I read that. *lol* You are dumb like shit!
>
> A SET OF WORDS contains "repetitions", are you SERIOUS, OR JUST AN ASSHOLE FULL OF SHIT?

WM might plausibly have had in mind, e.g., a, aa, aaa, aaaa, ad inf.
supposing that 'a' is a word in the Bible. The set of them is of
infinite cardinality and would serve very well as |N.

[Alan Smaill]

WM <wolfgang.m...@hs-augsburg.de> writes:

> No, the Peano axioms supply the set of words of the Bible and their
> repetitions. These have no numerical value and have nothing to do with
> mathematics. They supply nothing of number theory. Nevertheless the
> rules of set theory and logic do not deny or exclude that model.

In fact, Peano's axioms are Correct According To WM, when understood
as assertions about a potentially infinite collection.

Isn't that so?


>
> Regards, WM

--
Alan Smaill

[Alan Smaill]

Dan Christensen <Dan_Chr...@sympatico.ca> writes:

> On Saturday, July 9, 2016 at 9:00:00 AM UTC-4, Peter Percival wrote:
>> Dan Christensen wrote:
>
>> >
>> > Do you agree that Peano's Axioms for (N,S,0) gives us the essential
>> > properties of the set of natural numbers to the extent that all of
>> > modern number theory can be derived from them using nothing more than
>> > the ordinary rules of logic and set theory?
>>
>> I'm not sure what "all of modern number theory" is. Naturally, I am
>> worried about incompleteness.
>
> I doubt that number theorists have even given it a second thought.

The following is from Terry Tao (look him up in connection
with number theory):


"So, if Goldbach's conjecture is undecidable for some given axiom
system, what this would imply is that every "true" even natural number
larger than 4 is the sum of two primes (otherwise we could disprove
Goldbach with an argument of finite length), but that there exists a
more exotic number system (larger than the true natural numbers)
obeying those axioms which contains some even exotic number that is
not the sum of two (exotic) primes. (Note that the length of a proof
has to be true natural number, rather than an exotic one, so the
existence of an exotic counterexample cannot be directly converted to
a disproof of Goldbach.) This is an unlikely scenario, but not an a
priori impossible one (as one can see from the example of Goodstein's
theorem or the Paris-Harrington theorem)."

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

--
Alan Smaill

[Dan Christensen]

You yourself are an example, Mucke. Though I'm sure you still don't understand how and will still deny it, you have unknowingly accepted one of Peano's Axiom (induction) and have rejected the rest. As a result, you cannot even prove the existence of a single number. Accepting Peano's first axiom (1 in N) would have solved that problem.

[WM]

Am Sonntag, 10. Juli 2016 15:26:46 UTC+2 schrieb Dan Christensen:


> you have unknowingly accepted one of Peano's Axiom

No.

> (induction)

of numbers, not of "successors".

> and have rejected the rest.

Why should I remedy a mistake that I have not committed?

> As a result, you cannot even prove the existence of a single number. Accepting Peano's first axiom (1 in N) would have solved that problem.

So you cannot even read simplest text like 1 in |N?

You are a very instructive example of an excellent matheologian.

Regads, WM

[WM]

Am Sonntag, 10. Juli 2016 12:08:33 UTC+2 schrieb Me:


> We might consider the set B = {"I", "II", "III", IIII", ...} for example. Where the alphabet just consists of "I".

That is repetition. It can be applied to words of the Bible too.

>
> > They are sufficient. But in order to use them as a denotation of |N, you
> > must first know |N.
>
> Huh? All we need is a MODEL for the Peano axioms:

No, mathematics needs more.

>
> > The Peano axioms don't supply that meaning, in particular they do not
> > prove the constant distance.
>

> We can PROVE from that Peano axioms that
>
> for every n e IN:
> there is no x e IN such that n < x < S(n) ,

No. 1, 2, 3/2, 3, 4, ....

n = 1 < 3/2 < 2 = S(1).

>
> after defining < the usual way.

It has been defined the usual way for the above inequality.

With other words, we can PROVE that for any element n in IN there is no element in IN such that this element is "between" n and S(n). Got that?

No.

>
> Moreover we can PROVE that for every n e IN:
>
> S(n) - n = 1 ,
>
> after defining + (and then -) the usual way.

That way is not defined in he Peano axioms.

> With other words, we can PROVE that the "distance" between any element n in IN and its successor S(n) is constant (and equals to 1). Got that?

Not from the Peano axioms. Try to learn some arithmetic.

>
> Maybe you prefer the following form:
>
> An e IN Am e IN: S(n) - n = S(m) - m ,
>
> the "distance" between S(x) and x is "constant".

But that is not defined by Peano.

Regards, WM

[Jim Burns]

On 7/10/2016 12:57 AM, Dan Christensen wrote:
> On Saturday, July 9, 2016 at 12:31:32 PM UTC-4, Jim Burns wrote:
>> On 7/9/2016 8:20 AM, Dan Christensen wrote:
>> [to Peter Percival ]
>>>> Dan Christensen wrote:

>>>>> How about simply, the essential properties of the
>>>>> natural numbers? Too definition-like for you? Oh, well.
>>>
>>> Do you agree that Peano's Axioms for (N,S,0) gives us
>>> the essential properties of the set of natural numbers
>>> to the extent that all of modern number theory can be derived
>>> from them using nothing more than the ordinary rules of
>>> logic and set theory?
>>
>> The Peano axioms give us what they give us. If those are
>> the properties that you want the natural numbers to have,
>> then I'd say it's at least not terrible.
>
> Evasive.

Your question is vague, so that is the best answer I have.
However, I see below that you want a less-than-best
answer, so I'll give you one -- below.

>> (There was this poster once-upon-a-time who defined
>> the natural numbers as pretty much any infinite set,
>> and the _familiar_ +, *, < as pretty much anything.
>> That, I would call terrible.)
>>
>> If you want to argue from the essential-ness of those properties
>> that PA _should be_ taken as defining the natural numbers,
>> then you should say what you mean by "essential" in a
>> non-circular way -- and then give your argument.
>
> Each of Peano's Axioms is essential in the sense that
> removing or negating it would not allow us to derive
> modern number theory as we know it.
> Now please answer the above question.

No, I disagree. I would have liked you to define "essential
property", but you're not going to. So, I will, and
the way I do, you're wrong.

Quite a lot of number theory is done with first order
theories, which excludes a second order induction axiom.

Now, it might be that you would like to exclude the work
done with first order theories from "number theory".
You've also at least hinted that sentences provably
true-but-not-provable are not what you mean by
"essential properties".

You could say what you mean by "essential property"
and then show that these things are not essential properties,
as you mean the term. You haven't done that --
instead you called me "evasive" for asking you to
make yourself clear.

Or, you could define "essential property" so as to give
you the result that you so clearly want: the Peano
axioms and no other are what are needed to describe
natural numbers. Since you are so set against doing the
first, I'm pretty confident you're doing this.

_However_, if you define "essential properties" of natural
numbers so that they are exactly the properties implied by the
Peano axioms, then, _yes_ , exactly the Peano axioms are needed
to imply all and only the essential properties of the natural
numbers. Of course, this is circular and not very
interesting.

What I strongly suspect is that you want to accept or
reject properties nominally on the basis of they're
being (vaguely) "essential" or "non-essential" but
_in fact_ on the basis of they're being Peano axioms
or theorems of Peano axioms. SO, I strongly suspect
you of _equivocating_ about what you mean by "essential",
the way you did about what you mean by 2 earlier.

Of course, if you were to say what you mean by "essential"
and argue that those sort of properties were needed
(and only those), my suspicions would be shown to be
groundless.


On a separate note, please stop evading my point
(made with your help) about corresponding elements
of isomorphic models *NOT* being identical.

<unsnip>

> And that any theorem derived for a structure (N',S',0')
> that is order-isomorphic to (N,S,0) can be translated to
> a theorem in (N,S,0) simply be substituting N for N',
> S for S' and 0 for 0'?

Do you agree that _does not_ mean this?
S(S(0)) = S'(S'(0')) [??]

Rather, it _does_ mean, for the isomorphism g: N -> N'
g( S(S(0)) ) = S'(S'(0'))

Then, for the two structures
( { 1, 2, 3, ... }, 1, x |-> x+1 )
and
( { 1, 1/2, 1/4, ... }, 1, x |-> x/2 )
which both satisfy the Peano axioms, there exists an
isomorphism g, where g(1) = 1 and g(x+1) = g(x)/2

Then we would have g(2) = 1/2 -- which does not mean 2 = 1/2 .

If you want to say that the Peano axioms define the
natural numbers _up to isomorphism_ then I would agree.
What does "define _up to isomorphism_ " mean?

Consider the _numerals_ for the natural numbers.
In base ten, we have 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
In base two, we have 0, 1, 10, 11, 100, 101, 110, 111, ...
They are both perfectly fine ways (among many) to label
natural numbers. _There is no one way_ to do that.
In one context, S(S(0)) = 2 .
In another context, S(S(0)) = 10 .
This does not mean (the numeral) 2 = (the numeral) 10 .

There is an isomorphism between decimal numerals and
binary numerals that preserves everything we care about
as we change from one labeling scheme to another. So,
whether we are using decimal, binary, or some other
kind of numerals to describe natural numbers is something
that just doesn't get much attention. You pick what
numeral system you want, and I'll pick whatever numeral
system I want, and if they're different, we'll translate
back and forth. So, which we use doesn't get attention because
it really doesn't deserve attention.

In the same way, which isomorphic model of PA we are talking
about really doesn't deserve attention (and all models of PA
are isomorphic). They are all equally good, and we can translate
back and forth as needed.

I should point out that, as reasonable as this sounds, it
leads us to calling 1/2 a natural number _up to isomorphism_
because models of PA exist in which 1/2 plays the role of
a natural number. You, Dan, have been maybe a little resistant
to this conclusion.

If you have some other idea as to what it means to be a
natural number, _period_ and not _up to isomorphism_ ,
I haven't seen it from you yet. Ridiculing the idea of
1/2 being a natural number says what you think it is _not_ .
It doesn't say what it _is_ , period.

</unsnip>

[WM]

No they are trash, unable to define the natural numbers. But they are dangerous too because they mislead weak thinkers like DC or Franz Fritsche to utter complete nonsense. They should be dumped with a big warning sign "exhumation prohibited".

Regards, WM

[WM]

Am Sonntag, 10. Juli 2016 11:42:31 UTC+2 schrieb Me:
> On Saturday, July 9, 2016 at 11:53:35 PM UTC+2, Peter Percival wrote:
> > Me wrote:
> > > On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:
> > >
> > >> No, the Peano axioms supply the set of words of the Bible and their
> > >> repetitions.
> > >
> > > No, idiot, the don't "supply" anything. Moreover, since the set of words in the Bible is FINITE, it can't be used as a denotation of "IN" in the Peano axioms.
> >
> > "and their repetitions".
>

> Yeah, idiot, I read that but I am dumb like shit!
>
> A SET OF WORDS contains "repetitions"? I believe that I AM AN ASSHOLE FULL OF SHIT.

Don't worry too much. Even creatures like you have a right to live.

Regards, WM

[Me]

On Sunday, July 10, 2016 at 3:56:30 PM UTC+2, WM wrote:
> Am Sonntag, 10. Juli 2016 12:08:33 UTC+2 schrieb Me:
> >
> > We might consider the set B = {"I", "II", "III", IIII", ...} for
> > example. Where the alphabet just consists of "I".
> >

> [...] It can be applied to words of the Bible too.

If we consider the words in the (King James) bible to be our "alphabet" and then build up our new words based on that "alphabet", we get an infinite set of "words" (containing words like "EveAdamEve", etc.).

In addition to this we have to select a "first element" in this set and to supply a "succssor function" on this set.

Well, this can be done easily by using a so called "lexicographic order". Then we have 1 := "a" and let S(X) the word following X (in the lexicographic order).

> > All we need is a MODEL for the Peano axioms:
> >
> No, mathematics needs more.

You know SHIT about mathematics, so please shut up, idiot.

> > > The Peano axioms don't supply that meaning, in particular they do not
> > > prove the constant distance.
> > >

> > We can PROVE from the Peano axioms that

> >
> > for every n e IN:
> > there is no x e IN such that n < x < S(n) ,
> >
> No.

Yes, idiot.

> > after defining < the usual way. < <
> >
> It has been defined the usual way for the above inequality.

Great. Then you should be able to PROVE that theorem yourself, moron.

> With other words, we can PROVE that for any element n in IN there is no
> element in IN such that this element is "between" n and S(n). Got that?
>
> No.

Sure, WE can. Maybe you can't, I guess.

> > Moreover we can PROVE that for every n e IN:
> >
> > S(n) - n = 1 ,
> >
> > after defining + (and then -) the usual way.
> >
> That way is not defined in he Peano axioms.

Indeed, the axioms are AXIOMS, man, NOT definitions. Who'd have thunk it?

The DEFINTIONS must be stated by US. Following the Peano axioms.

Hint: "The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in second-order logic, and are shown to be unique using the Peano axioms."

https://en.wikipedia.org/wiki/Peano_axioms#Addition

> > With other words, we can PROVE that the "distance" between any element
> > n in IN and its successor S(n) is constant (and equals to 1). Got that?
> >
> Not from the Peano axioms.

You are completely screwed up, man.

> > Maybe you prefer the following form:
> >
> > An e IN Am e IN: S(n) - n = S(m) - m ,
> >
> > the "distance" between S(x) and x is "constant".
> >
> But that is not defined by Peano.

Actually, he DID define addition the USUAL WAY after stating the axioms.

But right, they are not part of the (second order) Peano axioms. No one ever claimed, they are.

[Virgil]

In article <643ffa3c-e5f8-4548...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> > You are losing it here,
>
> No I gain an incredible example.

In Mathematics, Cantor has been right in
claiming Card(Q) = Card(N) and Card(N) < Card(R).

So Card(Q) < Card(R), with fewer rationals than reals

In WM's witless worthless wacky world of WMytheology,
WM falsely claims Cantor is twice wrong,
WM claiming Card(Q) > Card(N) and Card(N) = Card(R)
So in WM's witless worthless wacky world of WMytheology
WM claims Card(Q) > Card(R), with more rationals than reals.

But until WM can produce some of his mythical unreal rationals,
Cantor remains right and WM remains wrong!

[Virgil]

In article <33b2a8f0-c833-4a70...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Sonntag, 10. Juli 2016 06:57:41 UTC+2 schrieb Dan Christensen:
>
>
> > Each of Peano's Axioms is essential in the sense that removing or negating
> > it would not allow us to derive modern number theory as we know it.
>

> Of course all axioms are required.

> But the natural numbers are not defined by his axioms.

Inductive sets are defined by those axioms, of which the set of natural
numbers is the most familiar. All such inductive sets are
order-isomorphic under the order defined by successorship. And that
order-isomorphism is all that is requires for induction.

----------------------------------------

In Mathematics, Cantor is provably right,

claiming Card(Q) = Card(N) and Card(N) < Card(R).
So Card(Q) < Card(R), with fewer rationals than reals

In WM's witless worthless wacky world of WMytheology,

WM claims Cantor is twice wrong,

WM claiming Card(Q) > Card(N) and Card(N) = Card(R)
So in WM's witless worthless wacky world of WMytheology
WM claims Card(Q) > Card(R), with more rationals than reals.

But until WM can produce some unreal rationals,

[Virgil]

In article <770f0ffc-4148-4395...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Sonntag, 10. Juli 2016 15:26:46 UTC+2 schrieb Dan Christensen:
>
>
> > you have unknowingly accepted one of Peano's Axiom
>
> No.

Without Peano's axioms, or others just as good, there is no such thing
as an inductive argument possible.

But in WM's witless worthless wacky world of WMytheology that is not a
problem because no other valid forms of argument allowed there anyway.

In Mathematics, Cantor is provably right,

Card(Q) = Card(N) and Card(N) < Card(R)

So Card(Q) < Card(R), with fewer rationals than reals

In WMaths/WMytheology, WM claims Cantor is wrong,

WM claims Card(Q) > Card(N) and Card(N) = Card(R)
So Card(Q) > Card(R), with more rationals than reals.

[Virgil]

In article <572f2f4d-4c67-490b...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Samstag, 9. Juli 2016 23:38:37 UTC+2 schrieb Me:
> > On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:
> >
> > > No, the Peano axioms supply the set of words of the Bible and their
> > > repetitions.
> >
> > since the set of words in the Bible is FINITE, it can't be used as a
> > denotation of "IN" in the Peano axioms.
>
> I pointed to their infinite repetitions, combinations, or concatenations.
> They are sufficient. But in order to use them as a denotation of |N, you must
> first know |N.

Thus in WMytheology to learn |N one must first know |N?

Then when in WMytheology one can never properly learn |N, and clearly WM
never has learnt it properly. According to WM there must be both a last
member of |N and no last member of |N, which is nonsense.

> The Peano axioms don't supply that meaning, in particular they
> do not prove the constant distance.

Because nothing in the inductive sets defined by the Peano axioms
requires a constant distance between members.

The Peano axioms define inductive sets, infinite well-ordered sets with
unique non-successor, of which the set of naturals is only one example.

In Mathematics, Cantor is provably right,

claiming Card(Q) = Card(N) and Card(N) < Card(R).

So Card(Q) < Card(R), with fewer rationals than reals

In WM's witless worthless wacky world of WMytheology,
WM claims Cantor is twice wrong,
WM claiming Card(Q) > Card(N) and Card(N) = Card(R)
So in WM's witless worthless wacky world of WMytheology
WM claims Card(Q) > Card(R), with more rationals than reals.

But until WM can produce some unreal rationals,
Cantor remains right and WM remains wrong!

[Virgil]

In article <059649af-3e8f-4a6f...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Sonntag, 10. Juli 2016 12:08:33 UTC+2 schrieb Me:
>
>
> > We might consider the set B = {"I", "II", "III", IIII", ...} for example.
> > Where the alphabet just consists of "I".
>
> That is repetition. It can be applied to words of the Bible too.

But the words of the Bible do not form an inducive set (an infinite
well-ordered set with one and only one non-successor).

> >
> > > They are sufficient. But in order to use them as a denotation of |N, you
> > > must first know |N.
> >
> > Huh? All we need is a MODEL for the Peano axioms:
>
> No, mathematics needs more.

WM may need more, but no one else does!

> >
> > > The Peano axioms don't supply that meaning, in particular they do not
> > > prove the constant distance.

Peano sets do not require constant distance between one member and the
next. They only require an infinite well-ordered set with one and only
one non-successor member.

Any such infinite well-ordered set with one and only one non-successor
member is a Peano set since it must have a first member, the
non-successor, and for each member a unique successor member!

> >
> > We can PROVE from that Peano axioms that
> >
> > for every n e IN:
> > there is no x e IN such that n < x < S(n) ,
>
> No. 1, 2, 3/2, 3, 4, ....

If |N = {1, 2, 3/2, 3, 4, ....} then S(2) = 3/2 and S(3/2) = 3

> With other words, we can PROVE that for any element n in IN there is no
> element in IN such that this element is "between" n and S(n). Got that?

Maybe in WM's witless worthless wacky world of WMytheology but NOT in
any Peano set.

[Virgil]

In article <cdd261cf-961a-4e81...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Sonntag, 10. Juli 2016 14:40:06 UTC+2 schrieb Alan Smaill:
> > WM <wolfgang.m...@hs-augsburg.de> writes:
> >
> > > No, the Peano axioms supply the set of words of the Bible and their
> > > repetitions. These have no numerical value and have nothing to do with
> > > mathematics. They supply nothing of number theory. Nevertheless the
> > > rules of set theory and logic do not deny or exclude that model.
> >
> > In fact, Peano's axioms are Correct According To WM, when understood
> > as assertions about a potentially infinite collection.
> >
> > Isn't that so?
>
> No they are trash, unable to define the natural numbers.

The Peano axioms define a class of ordered sets each order-isomorphic to
the ordered set of natural numbers

But they are dangerous too, for weak thinkers like WM, because they
mislead such weak thinkers as WM into claiming complete nonsense.

In Mathematics, Cantor is provably right,
claiming Card(Q) = Card(N) and Card(N) < Card(R).
So Card(Q) < Card(R), with fewer rationals than reals

In WM's witless worthless wacky world of WMytheology,
WM claims Cantor is twice wrong,
WM claiming Card(Q) > Card(N) and Card(N) = Card(R)

So in WM's witless worthless wacky world of WMytheology
WM also claims Card(Q) > Card(R), with more rationals than reals.

But until WM can produce some unreal rationals,
Cantor remains right and

WM is
WRONG !
AGAIN ! !
AS USUAL ! ! !

[Virgil]

On Saturday, July 9, 2016 at 3:14:42 PM UTC+2, WM wrote:

> No, the Peano axioms supply the set of words of the Bible and their
> repetitions.

In Peano-axiom supplied sets there are no repetitions of members in any
one set and far more members in any one such set than the Bible can
supply as words.

So WM is
LYING !

[Virgil]

In article <2362181e-6902-4a5f...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Sonntag, 10. Juli 2016 02:56:57 UTC+2 schrieb Virgil:
>
>
> > > That structure is given by counting +1 in a much better form.

Nothing WM gives is in any better form than garbage.

[WM]

Am Sonntag, 10. Juli 2016 17:29:57 UTC+2 schrieb Me:


> > But that is not defined by Peano.
>
> Actually, he DID define addition the USUAL WAY after stating the axioms.
>
> But right, they are not part of the (second order) Peano axioms. No one ever claimed, they are.

You are wrong. We have a living example of the contrary.

Regards, WM

[WM]

Am Sonntag, 10. Juli 2016 17:29:57 UTC+2 schrieb Me:

> > > All we need is a MODEL for the Peano axioms:
> > >
> > No, mathematics needs more.
>

> I know SHIT about mathematics.

>
> > > > The Peano axioms don't supply that meaning, in particular they do not
> > > > prove the constant distance.
> > > >
> > > We can PROVE from the Peano axioms that
> > >
> > > for every n e IN:
> > > there is no x e IN such that n < x < S(n) ,

> > No. 1, 2, 3/2, 3, 4, ....

> I see, I am an idiot.

>

> > > after defining + (and then -) the usual way.
> > >
> > That way is not defined in he Peano axioms.
>
> Indeed, the axioms are AXIOMS, man, NOT definitions.

Axioms are often definitions. 1 is a natural number is a definition.

> > > With other words, we can PROVE that the "distance" between any element
> > > n in IN and its successor S(n) is constant (and equals to 1). Got that?
> > >
> > Not from the Peano axioms.
>

> I was completely screwed up.

You are welcome.

Regards, WM

[Virgil]

In article <05e511c8-0f1a-4a03...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Sonntag, 10. Juli 2016 17:29:57 UTC+2 schrieb Me:
>
>
> > > > All we need is a MODEL for the Peano axioms:
> > > >
> > > No, mathematics needs more.
> >
> > I know SHIT about mathematics.
> >
> > > > > The Peano axioms don't supply that meaning, in particular they do not
> > > > > prove the constant distance.
> > > > >
> > > > We can PROVE from the Peano axioms that
> > > >
> > > > for every n e IN:
> > > > there is no x e IN such that n < x < S(n) ,
>
> > > No. 1, 2, 3/2, 3, 4, ....
>
> > I see, I am an idiot.
> >
>
> > > > after defining + (and then -) the usual way.
> > > >
> > > That way is not defined in he Peano axioms.
> >
> > Indeed, the axioms are AXIOMS, man, NOT definitions.
>
> Axioms are often definitions.

But the axioms of WM's witless worthless wacky world of WMytheology are
not the axioms of proper mathematics.

In Proper Mathematics, Cantor is provably right,

claiming Card(Q) = Card(N) and Card(N) < Card(R)

So in Mathematics Card(Q) < Card(R), with fewer rationals than reals.

In his WMytheology, WM claims Cantor is wrong,
there WM claims Card(Q) > Card(N) and Card(N) = Card(R)
So in WM's witless worthless wacky world of WMytheology,

Card(Q) > Card(R), with more rationals than reals.

In MY math, like Cantor's, there are far more reals than rationals

So are there more rationals than reals or more reals than rationals in
your mathematics?

[Virgil]

In article <ab3b2aca-4882-4f17...@googlegroups.com>,
WM <wolfgang.m...@hs-augsburg.de> wrote:


> You are wrong.

It is WM who is wrong!

In Mathematics, Cantor is provably right,

Card(Q) = Card(N) and Card(N) < Card(R)

So Card(Q) < Card(R), with fewer rationals than reals

In WMytheology, WM claims Cantor is twice wrong,

WM claims Card(Q) > Card(N) and Card(N) = Card(R)

So Card(Q) > Card(R), with WM claiming more rationals than reals.

So who is the one who is wrong?

[Dan Christensen]

On Sunday, July 10, 2016 at 9:44:59 AM UTC-4, WM wrote:
> Am Sonntag, 10. Juli 2016 15:26:46 UTC+2 schrieb Dan Christensen:
>
>
> > you have unknowingly accepted one of Peano's Axiom
>
> No.
>
> > (induction)
>
> of numbers, not of "successors".
>
> > and have rejected the rest.
>
> Why should I remedy a mistake that I have not committed?
>

It was a HUGE mistake, Mucke. You are looking like a complete idiot here.

> > As a result, you cannot even prove the existence of a single number. Accepting Peano's first axiom (1 in N) would have solved that problem.
>
> So you cannot even read simplest text like 1 in |N?
>

I can indeed, and it was NOT included in your "axioms," Mucke. Neither is:

- For in N: S(x) in N

- For all x in N: (S(x)=/=1

- For x, y in N: [S(x)=S(y) => x=y]


You can prove neither of these results in your goofy number system, Mucke. And without either of them your goofy number system is absolutely useless.

[Dan Christensen]

On Sunday, July 10, 2016 at 9:58:45 AM UTC-4, Jim Burns wrote:
> On 7/10/2016 12:57 AM, Dan Christensen wrote:

> >>> Do you agree that Peano's Axioms for (N,S,0) gives us
> >>> the essential properties of the set of natural numbers
> >>> to the extent that all of modern number theory can be derived
> >>> from them using nothing more than the ordinary rules of
> >>> logic and set theory?
> >>
> >> The Peano axioms give us what they give us. If those are
> >> the properties that you want the natural numbers to have,
> >> then I'd say it's at least not terrible.
> >
> > Evasive.
>
> Your question is vague,

You have painted yourself into a corner, Jim. Your insistence that {1, 1/2, 1/4, ...} and the set of natural numbers are really "the same thing" is going nowhere. Neither is your denial that the Peano Axioms define the set of natural numbers -- not in the context of formally deriving number theory anyway.

[snip]

>
> Of course, if you were to say what you mean by "essential"
> and argue that those sort of properties were needed
> (and only those), my suspicions would be shown to be
> groundless.
>

That's pretty much what I say, isn't it?

>
> On a separate note, please stop evading my point
> (made with your help) about corresponding elements
> of isomorphic models *NOT* being identical.
>

[snip]

So, you finally admit that {1, 1/2, 1/4, ...} and the set of natural numbers are NOT really "the same thing" after all. Good for you, Jim!


Maybe this will help with the rest: In number theory we don't go around proving what is or is not a number. That is uninteresting and purely the realm of navel gazers and cranks these days. We take as given the set of natural of numbers as defined by Peano's Axioms (i.e. their 5 essential properties) and set out to establish other of their properties using only the rules of ordinary logic and set theory.

[Dan Christensen]

On Saturday, July 9, 2016 at 9:14:42 AM UTC-4, WM wrote:

> Am Samstag, 9. Juli 2016 14:20:12 UTC+2 schrieb Dan Christensen:
>
> >
> > Do you agree that Peano's Axioms for (N,S,0) gives us the essential properties of the set of natural numbers to the extent that all of modern number theory can be derived from them using nothing more than the ordinary rules of logic and set theory?
>

> No, the Peano axioms supply the set of words of the Bible and their repetitions. These have no numerical value and have nothing to do with mathematics.

This from an idiot who mistakenly took a statement of Peano's induction axiom for the definition of the natural numbers while rejecting the other axioms. Pathetic.

> They supply nothing of number theory.

No, it is your goofy number system that supplies nothing, not even the most elementary results of basic arithmetic -- not 2+2=4, 1=/=2, or even the existence a single number! Truly a dead end.

Peano's Axioms, on the other can be used to derive all of modern number theory. You must be so jealous!


**********

More absurd quotes from Wolfgang Muckenheim (WM):

“In my system, two different numbers can have the same value.”
-- sci.math, 2014/10/16

“1+2 and 2+1 are different numbers.”
-- sci.math, 2014/10/20

“1/9 has no decimal representation.”
-- sci.math, 2015/09/22

"0.999... is not 1."
-- sci.logic 2015/11/25

“Axioms are rubbish!”
-- sci.math, 2014/11/19

“No set is countable, not even |N.”
-- sci.logic, 2015/08/05

“Countable is an inconsistent notion.”
-- sci.math, 2015/12/05

“A [natural] number with aleph_0 digits is not less than aleph_0.”
-- sci.math, 2015/08/12

“The notion of aleph_0 is not meaningful.”
-- sci.math, 2015/08/28


Slipping ever more deeply into madness...

“There is no actually infinite set |N.”
-- sci.math, 2015/10/26

“|N is not covered by the set of natural numbers.”
-- sci.math, 2015/10/26

“The set of all rationals can be shown not to exist.”
--sci.math, 2015/11/28

“Everything is in the list of everything and therefore everything belongs to a not uncountable set.”
-- sci.math, 2015/11/30

"'Not equal' and 'equal can mean the same.”
-- sci.math, 2016/06/09



A special word of caution to students: Do not attempt to use WM's “system” (MuckeMath) in any course work in any high school, college or university on the planet. You will fail miserably. MuckeMath is certainly no shortcut to success in mathematics.

Using WM's “axioms” for the natural numbers, he cannot prove that 1=/=2 or the existence of even a single number. It is truly a dead-end.

[Me]

On Sunday, July 10, 2016 at 11:41:28 PM UTC+2, WM wrote:
> Am Sonntag, 10. Juli 2016 17:29:57 UTC+2 schrieb Me:
> >

> > Actually, Peano defined addition the USUAL WAY after stating the axioms.

> >
> > But right, they are not part of the (second order) Peano axioms. No one
> > ever claimed, they are.
> >
> You are wrong. We have a living example of the contrary.

I don't think so.

[Me]

On Sunday, July 10, 2016 at 11:47:44 PM UTC+2, WM wrote:
>
> I know SHIT about mathematics.

Yes, we know that already, man.

> I see, I am an idiot.

Agree.

> > axioms are AXIOMS, man, NOT definitions.
> >
> Axioms are often definitions.

Man, you are totally screwed up, really. Dumb, igorant and uninformed.

Hint: No, AXIOM aren't definitions. Never. That's why the are called AXIOMS, *not* DEFINITIONS. (Is there ANY mathematical concept you are not too dumb to comprehend?)

> '1 is a natural number' is a definition.

No, idiot. This is (or might be) an AXIOM.

Mückenheim, you really should ask your superiors for some vacation.

Hint: If you want to introduce a symbol with a definition you have first to establish (by a proof) "existince" and "uniqueness". For example

0 is the smallest natural number ,

might be a definition (in a certain system). On the other hand the statement

0 is a natural number ,

doesn't define ANYTHING. (With other words, it's not a proper definition.)

> I was completely screwed up.

I see.

[WM]

Am Montag, 11. Juli 2016 11:02:31 UTC+2 schrieb Me:


> Hint: No, AXIOM aren't definitions. Never.

> > '1 is a natural number' is a definition.
>

> No, This is (or might be) an AXIOM.

It is both and even more, namely a theorem and a contradiction of your above claim.

>
> Hint: If you want to introduce a symbol with a definition you have first to establish (by a proof) "existince" and "uniqueness".

Wrong. God has been introduced long before his existence was "proved" by Gödel. But also 1 has been introduced before its existence was proved.

The existence of neither of both, God and 1, can be proved. It is only your incapable mind (completely perverted by what you think is "modern logic") that lets you dive into nonsense and sympathize with fools who "prove" 1 + 1 = 2.

Regards, WM

Regards.

[Me]

On Monday, July 11, 2016 at 1:48:29 PM UTC+2, WM wrote:

> The existence of neither of both, God and 1, can be proved. It is only your
> incapable mind (completely perverted by what you think is "modern logic")
> that lets you dive into nonsense and sympathize with fools who "prove"
> 1 + 1 = 2.

Yeah, Mücke, it must be hard for you that you can't do that in your goofy number system that supplies nothing, not even the most elementary results of basic arithmetic -- not 1+1=2, 1=/=2, or even the existence a single number!

[Me]

On Monday, July 11, 2016 at 2:09:54 PM UTC+2, Me wrote:

> Yeah, Mücke, it must be hard for you that you can't do that in your goofy
> number system that supplies nothing, not even the most elementary results
> of basic arithmetic -- not 1+1=2, 1=/=2, or even the existence a single
> number!

Hint: Try to prove

IN =/= {}
or
Ex(x e IN)

from your idiotic "axiom system fo the natural numbers".

[Peter Percival]

How is it proved that 1 exists? One might have an axiom that says that
there is a set and 1 is an element of that set. The proof that 1 exists
is then one line long. But that's not very impressive, compare

there is an even number > 2 that is not the sum of two primes

as an axiom. Now we have a one-line proof that the Goldbach conjecture
is false.

Clearly (!) what we want is not just axioms, but axioms that are true,
so is it true that

there is a set and 1 is an element of that set

rather than it just being postulated?

So, it seems to me, the innocent-sounding claim that 1 exists raises
such difficult questions as

i) what does it mean to say that a mathematical object exists?

ii) what does it mean to say that a mathematical statement is true?

I will be happy to hear the answers.

--
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.
Ulysses, Alfred, Lord Tennyson

[Me]

On Monday, July 11, 2016 at 2:57:40 PM UTC+2, Peter Percival wrote:

> How is it proved that 1 exists.

Are you an drugs, man?

I refered to the "fools who 'prove' 1 + 1 = 2."

[Me]

On Monday, July 11, 2016 at 2:57:40 PM UTC+2, Peter Percival wrote:

> How is it proved that 1 exists?

You might read Quine's "On what there is" for a starter. Actually, the first question is "How can we even STATE that 1 exists?" (especially in the context of a formal language, say FOPL)? Quine proposes the following approach:

Ex(x = 1)
"1 exists."

Now starting with the Peano axiom

1 e IN

we can indeed prove

Ex(x e IN & x = 1)

and hence

Ex(x = 1).

> One might have an axiom that says that
> there is a set and 1 is an element of that set.

Again from the Peano axiom just mentioned, we get:

Ey(1 e y)

> But that's not very impressive,

Impressive or not, but it can be PROVED!

Btw, in Free Logic we actually have an "existence predicate" which might be defined the following way:

E!t := Ex(x = t)

(using Quines' idea). In Free Logic we might have terms /t/ such that ~E!t.

See:
https://en.wikipedia.org/wiki/Free_logic

> Clearly (!) what we want is not just axioms, but axioms that are true,

*lol* Meaning?

> so is it true that
>
> there is a set and 1 is an element of that set
>
> rather than it just being postulated?

Well. Depends on your account concerning "mathematical facts" I'd say.

See:
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism

> So, it seems to me, the innocent-sounding claim that 1 exists raises
> such difficult questions as
>
> i) what does it mean to say that a mathematical object exists?
>
> ii) what does it mean to say that a mathematical statement is true?
>
> I will be happy to hear the answers.

See, link above. Those are genuine problems of the philosphy of mathematics.

My personal stance is fictionalism, see:
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Fictionalism

"... when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as fictitious as "Sherlock Holmes lived at 221B Baker Street"—but both are true according to the relevant fictions."

[Peter Percival]

Well who'd have thought it, with you asking if I am on drugs, and
laughing out loud.

> My personal stance is fictionalism, see:
> https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Fictionalism
>
> "... when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as fictitious as "Sherlock Holmes lived at 221B Baker Street"—but both are true according to the relevant fictions."
>

[Me]

On Monday, July 11, 2016 at 4:05:08 PM UTC+2, Peter Percival wrote:

> Well who'd have thought it, with you asking if I am on drugs, and
> laughing out loud.

Actually, I'd prefer you asking questions independend of WM's idiotic babbling.

[Jim Burns]

On 7/10/2016 11:28 PM, Dan Christensen wrote:
> On Sunday, July 10, 2016 at 9:58:45 AM UTC-4, Jim Burns wrote:
>> On 7/10/2016 12:57 AM, Dan Christensen wrote:

>>>>> Do you agree that Peano's Axioms for (N,S,0) gives us
>>>>> the essential properties of the set of natural numbers
>>>>> to the extent that all of modern number theory can be derived
>>>>> from them using nothing more than the ordinary rules of
>>>>> logic and set theory?
>>>>
>>>> The Peano axioms give us what they give us. If those are
>>>> the properties that you want the natural numbers to have,
>>>> then I'd say it's at least not terrible.
>>>
>>> Evasive.
>>
>> Your question is vague,
>
> You have painted yourself into a corner, Jim.
> Your insistence that {1, 1/2, 1/4, ...} and
> the set of natural numbers are really "the same thing"
> is going nowhere.

If by "the same thing" you mean " { 1, 1/2, 1/4, ... }
is _isomorphic_ to any other model of PA", then that is
well-established and even you have agreed (though you
have snipped your agreement for some reason).

If by "the same thing" you mean " { 1, 1/2, 1/4, ... }
is _equal_ to any other model of PA", then I have never
said that. I'm glad to hear that is going nowhere.

Dan, could you quote what I wrote that you clearly _think_
means that? I would more than happy to clear that up for you.

And (one more time): *what is the set of natural numbers* ,
_period_ (to borrow your term) _not_ up to isomorphism?

Keep in mind that N is only the _name_ of the set of natural
numbers often used in stating the Peano axioms, and you have
no way of knowing _from the Peano axioms_ whether the set so
named is { 1, 1/2, 1/4, ... } or some other set.

> Neither is your denial that the Peano Axioms
> define the set of natural numbers -- not in the context
> of formally deriving number theory anyway.

I've let you define the natural numbers as whatever satisfies
the Peano axioms. _You_ insist that _only_ the Peano axioms
(plus logic and set theory) be used to define them. So
_that is what you mean_ by "the set of natural numbers".

The problem then is that you _deny_ that
( { 1, 1/2, 1/4, ... }, 1, x }-> x/2 }
which you agree satisfies the Peano axioms,
uses { 1, 1/2, 1/4, ... } as the set of natural numbers.

If I deny you your definition by pointing out a contradiction,
then I suppose I am denying it -- but really the ball is in
your court. You can either change your definition to exclude
{ 1, 1/2, 1/4, ... } (and all other models except one,
presumably,) or you can include { 1, 1/2, 1/4, ... } as one
version of the set of natural numbers.

(That version is just as capable of being used to
"formally derive number theory" as any other model of PA.
Agreed?)

> [snip]
>
>>
>> Of course, if you were to say what you mean by "essential"
>> and argue that those sort of properties were needed
>> (and only those), my suspicions would be shown to be
>> groundless.
>
> That's pretty much what I say, isn't it?

Is that what you say? Could you repeat what you mean by
"essential properties" here?

I don't mean repeat that essential properties are what
you can derive from the Peano axioms given logic and
set theory. That is what you want to _conclude_ , so
it can't be your _premise_ , unless you don't mind making
a circular argument.

>> On a separate note, please stop evading my point
>> (made with your help) about corresponding elements
>> of isomorphic models *NOT* being identical.
>
> [snip]

So, I take that to be a "no" from you on my request for
you to stop evading.

> So, you finally admit that {1, 1/2, 1/4, ...} and
> the set of natural numbers are NOT really "the same thing"
> after all. Good for you, Jim!

What is this set of natural numbers that you speak of, Dan?
No, seriously: what is N ?

( { 1, 1/2, 1/4, ... }, 1, x |-> x/2 )
and
( { 1, 2, 3, ... }, 1, x |-> x+1 )
are still isomorphic models of the Peano axioms, Dan.

> Maybe this will help with the rest:
> In number theory we don't go around proving what is or
> is not a number.

To summarize your position:
( { 1, 1/2, 1/4, ... }, 1, x |-> x/2 ) _does not_
include the set of natural numbers and you don't have
to prove it.

> That is uninteresting and purely
> the realm of navel gazers and cranks these days.

Whereas making unsupported claims and insisting that
everyone agree with them (or be a crank) is the realm
of the serious mathematician, in your so-humble opinion.

> We take as given the set of natural of numbers
> as defined by Peano's Axioms (i.e. their 5 essential properties)
> and set out to establish other of their properties
> using only the rules of ordinary logic and set theory.

And we can do that. But then, you just described
( { 1, 1/2, 1/4, ... {, 1, x |-> x/2 )
as something with the set of natural numbers.

[Virgil]

In article <0764d15c-e045-4883...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Montag, 11. Juli 2016 11:02:31 UTC+2 schrieb Me:
>
>
> > Hint: No, AXIOM aren't definitions. Never.
>
> > > '1 is a natural number' is a definition.
> >
> > No, This is (or might be) an AXIOM.
>
> It is both and even more, namely a theorem and a contradiction of your above
> claim.

If something is an axiom then it is, by definition, NOT a theorem, and
vice versa. So WM is twice wrong!

> >
> > Hint: If you want to introduce a symbol with a definition you have first to
> > establish (by a proof) "existince" and "uniqueness".
>
> Wrong. God has been introduced long before his existence was "proved"

I was not aware that any god's existence had yet been proved, at least
to to the satisfaction of any agnostic.
>
> The existence of neither or both, God and 1, can be proved.

Then it must be "neither" because it certainly is not "both".

[Peter Percival]

Virgil wrote:

> If something is an axiom then it is, by definition, NOT a theorem,

The sequence which consists of an axiom, and nothing else, is a proof of
itself. So every axiom is a theorem.

> and
> vice versa. So WM is twice wrong!

[Virgil]

In article <nm0n4n$crk$1...@news.albasani.net>,

Peter Percival <peterxp...@hotmail.com> wrote:

> Virgil wrote:
>
> > If something is an axiom then it is, by definition, NOT a theorem,
>
> The sequence which consists of an axiom, and nothing else, is a proof of
> itself. So every axiom is a theorem.

It is true because it is assumed, not because it has been proved as
following from something else, which is my definition of theorem.

[Jim Burns]

On 7/11/2016 1:56 PM, Virgil wrote:
[to WM <wolfgang.m...@hs-augsburg.de> ]

>
> If something is an axiom then it is, by definition,
> NOT a theorem, and vice versa.

For given axioms and rules of inference,
a theorem of those axioms and rules is the last statement of
any finite list of statements, where each statement is
either an axiom or follows from previous statements in the
list by one of the rules.

This means that every axiom is a theorem, because it would
have a proof: a finite list (the axiom itself) every statement
of which is either an axiom or etc.

It's not an interesting theorem, but it's a theorem.

[...]

>
> I was not aware that any god's existence had yet been proved,
> at least to to the satisfaction of any agnostic.

Perhaps you mean "to the satisfaction of any _former_
agnostic". Naturally, if a former agnostic had been
satisfied that a god exists, they would no longer be
an agnostic. This would make what you say true
_by definition_ of agnostic. I suspect you mean to
say something more interesting.

[Jim Burns]

On 7/11/2016 2:09 PM, Virgil wrote:
> In article <nm0n4n$crk$1...@news.albasani.net>,
> Peter Percival <peterxp...@hotmail.com> wrote:
>> Virgil wrote:

>>> If something is an axiom then it is, by definition,
>>> NOT a theorem,
>>
>> The sequence which consists of an axiom, and nothing else,
>> is a proof of itself. So every axiom is a theorem.
>
> It is true because it is assumed, not because it has
> been proved as following from something else,
> which is my definition of theorem.

Virgil: I hope that your long association with WM has not infected
you with Mueckenheimlichkeit. Please be careful.

Including the axioms as theorems is usually what we mean
by theorems. Maybe you should refer to non-trivial theorems,
though those might be seen instead as excluding short,
obvious (but not only one-line) proofs.

If we are talking about equivalent collections of axioms,
it simplifies saying "they have the same theorems"m
because of course some axiom for one collection would be
a theorem for the other collection.

Also, we accept the truth of the theorems on precisely
the same basis as the truth of the (non-trivial) theorems.
There doesn't seem to be a good reason to declare that
axioms are not theorems, beyond allowing us to state
that axioms are not theorems. The value of being able
to do this is not clear to me.

[Me]

On Monday, July 11, 2016 at 8:26:36 PM UTC+2, Jim Burns wrote: [...]

To make a long story short:

Let's consider the following axiom system:

Ax. 1 Bla

and the following theorem in this system:

Th. 1 Bla

(Formal) Proof:

(1) Bla by Ax. 1

qed.

[Peter Percival]

Virgil wrote:
> In article <nm0n4n$crk$1...@news.albasani.net>,
> Peter Percival <peterxp...@hotmail.com> wrote:
>
>> Virgil wrote:
>>
>>> If something is an axiom then it is, by definition, NOT a theorem,
>>
>> The sequence which consists of an axiom, and nothing else, is a proof of
>> itself. So every axiom is a theorem.
>
> It is true because it is assumed, not because it has been proved as
> following from something else, which is my definition of theorem.

Then your definition is not the usual one.

[Dan Christensen]

On Monday, July 11, 2016 at 11:46:30 AM UTC-4, Jim Burns wrote:
> On 7/10/2016 11:28 PM, Dan Christensen wrote:
> > On Sunday, July 10, 2016 at 9:58:45 AM UTC-4, Jim Burns wrote:
> >> On 7/10/2016 12:57 AM, Dan Christensen wrote:
>
> >>>>> Do you agree that Peano's Axioms for (N,S,0) gives us
> >>>>> the essential properties of the set of natural numbers
> >>>>> to the extent that all of modern number theory can be derived
> >>>>> from them using nothing more than the ordinary rules of
> >>>>> logic and set theory?
> >>>>
> >>>> The Peano axioms give us what they give us. If those are
> >>>> the properties that you want the natural numbers to have,
> >>>> then I'd say it's at least not terrible.
> >>>
> >>> Evasive.
> >>
> >> Your question is vague,
> >
> > You have painted yourself into a corner, Jim.
> > Your insistence that {1, 1/2, 1/4, ...} and
> > the set of natural numbers are really "the same thing"
> > is going nowhere.
>
> If by "the same thing" you mean " { 1, 1/2, 1/4, ... }
> is _isomorphic_ to any other model of PA", then that is
> well-established and even you have agreed (though you
> have snipped your agreement for some reason).
>

It was YOUR claim, Jim. From an exchange between us in the thread, "JG's failed attempt to prove 2+2=4":


DC: Gosh, I don't know. You insisting that {1, 1/2, 1/4, ...} is the set of natural numbers must look pretty damn kooky to most folks. JG or WM couldn't have done better!

JB: You: No, I call the Peano Axioms "The Definition of the Set of Natural Numbers." Looks like you're the one doing the insisting.

DC: A much easier sell than {1, 1/2, 1/4, ...} being the set of natural numbers. (Kook City!)

JB: They're the same thing, Dan.

[End quote]

"The same thing" has quite a clear meaning, synonymous with "identical."

Deny it if you like, Jim -- I don't really care. The important thing here is that, for the purpose of formally developing number theory, Peano's Axioms along with the ordinary rules of logic and set theory have been shown to be quite sufficient as a starting point. Each of these axioms is a truly self-evident property of the natural numbers, and no others have been found to be necessary. In that sense Peano's Axioms define the set of natural numbers.

I don't find it particularly interesting that you can rename all the numbers in some way and still get the same results (with different names, of course). Maybe you can add that as an axiom. I wouldn't bother.

[Peter Percival]

So far as it is understood that the "(1) " and " by Ax. 1" are not
truly parts of the proof. The sequence

Bla

(just that) is the proof.

[Jim Burns]

On 7/11/2016 2:41 PM, Dan Christensen wrote:
> On Monday, July 11, 2016 at 11:46:30 AM UTC-4, Jim Burns wrote:
>> On 7/10/2016 11:28 PM, Dan Christensen wrote:
>> > On Sunday, July 10, 2016 at 9:58:45 AM UTC-4, Jim Burns wrote:
>> >> On 7/10/2016 12:57 AM, Dan Christensen wrote:

>> >>>>> Do you agree that Peano's Axioms for (N,S,0) gives us
>> >>>>> the essential properties of the set of natural numbers
>> >>>>> to the extent that all of modern number theory can be

>> >>>>> rules of from them using nothing more than the ordinary

>> >>>>> rules of logic and set theory?
>> >>>>
>> >>>> The Peano axioms give us what they give us. If those are
>> >>>> the properties that you want the natural numbers to have,
>> >>>> then I'd say it's at least not terrible.
>> >>>
>> >>> Evasive.
>> >>
>> >> Your question is vague,
>> >
>> > You have painted yourself into a corner, Jim.
>> > Your insistence that {1, 1/2, 1/4, ...} and
>> > the set of natural numbers are really "the same thing"
>> > is going nowhere.
>>
>> If by "the same thing" you mean " { 1, 1/2, 1/4, ... }
>> is _isomorphic_ to any other model of PA", then that is
>> well-established and even you have agreed (though you
>> have snipped your agreement for some reason).
>
> It was YOUR claim, Jim.

No, that is NOT what I claimed. And I saw that it could be
misunderstood after I read what I'd posted and posted a
clarification 13 minutes later.[1]

I claimed that two _statements_ , one of which you claim,
the other of which you deny, are the same thing.
And I told you that was what I mean 13 minutes later.

So, you're wrong, but _even if you were right_ and
I said { 1, 1/2, 1/4, ... } was "the same thing" as
the natural numbers, you are _still_ saying that the
Peano axioms define the natural numbers *AND* that
this set that satisfies the Peano axioms are not the
Peano axioms. Even if you were _right_ , it wouldn't
let you off the hook for your own errors.

> From an exchange between us in the thread,
> "JG's failed attempt to prove 2+2=4":
>
> DC:
> Gosh, I don't know. You insisting that {1, 1/2, 1/4, ...}
> is the set of natural numbers must look pretty damn kooky to
> most folks. JG or WM couldn't have done better!
>
> JB:
> You: No, I call the Peano Axioms
> "The Definition of the Set of Natural Numbers."
> Looks like you're the one doing the insisting.
>
> DC: A much easier sell than {1, 1/2, 1/4, ...}
> being the set of natural numbers. (Kook City!)
>
> JB:
> They're the same thing, Dan.
>
> [End quote]
>
> "The same thing" has quite a clear meaning, synonymous with
> "identical."

[1]
<quote me>
Clarifying:
The _statements_

> No, I call the Peano Axioms
> "The Definition of the Set of Natural Numbers."

and
{1, 1/2, 1/4, ...} is defined to be the natural numbers
(for S(x) = x/2 )
are the same thing.[1]
</quote me>
<https://groups.google.com/d/msg/sci.math/8LcBKHV28r4/7qs7ERfLBgAJ>

>
> Deny it if you like, Jim -- I don't really care.

No doubt you mean that you don't care about the truth of what
you say in the most non-kookiest way imaginable.

> The important
> thing here is that, for the purpose of formally developing
> number theory, Peano's Axioms along with the ordinary rules
> of logic and set theory have been shown to be quite sufficient
> as a starting point. Each of these axioms is a truly
> self-evident property of the natural numbers, and no others have
> been found to be necessary. In that sense Peano's Axioms define
> the set of natural numbers.

Then you should accept

( { 1, 1/2, 1/4, ... }, 1, x |-> x/2 )

as using the natural numbers. Do you?

> I don't find it particularly interesting that you can rename
> all the numbers in some way and still get the same results
> (with different names, of course). Maybe you can add that as
> an axiom. I wouldn't bother.

Wow. Poster boy for "unclear on the concept": Dan Christensen.

That doesn't take an axiom, Dan. It's just the result of
Peano's axioms having non-identical but isomorphic models.

All the statements of Peano's axioms and of anything that can
be derived from Peano's axioms can be interpreted to refer
to any model of Peano's axioms. If all you have are those
statements and their truth values _without interpretation_
there is _simply no way_ to distinguish one model from another
with only that information. Every model has the same information.

Isn't that your definition of the natural numbers, Dan?
_Your definition_ does not distinguish between different
(non-identical) models of the Peano axioms.

(Note: I emphasize _non-identical_ here because everyone
else understands that there are _always_ non-identical models
that cannot be distinguished by a theory that has any model
at all. It's only when there are _non-isomorphic_ model
that anyone typically talks about "different" models. This
is vary basic stuff that you are kicking a fuss up over,)

[Virgil]

In article <nm0p34$gmm$2...@news.albasani.net>,

Peter Percival <peterxp...@hotmail.com> wrote:

> Virgil wrote:
> > In article <nm0n4n$crk$1...@news.albasani.net>,
> > Peter Percival <peterxp...@hotmail.com> wrote:
> >
> >> Virgil wrote:
> >>
> >>> If something is an axiom then it is, by definition, NOT a theorem,
> >>
> >> The sequence which consists of an axiom, and nothing else, is a proof of
> >> itself. So every axiom is a theorem.
> >
> > It is true because it is assumed, not because it has been proved as
> > following from something else, which is my definition of theorem.
>
> Then your definition is not the usual one.

Theorem - Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Theorem?
In mathematics, a theorem is a statement that has been proven on the
basis of previously established statements, such as other theorems蟻nd
generally accepted statements, such as axioms. A theorem is a logical
consequence of the axioms.

Wiki seems to distinguish theorems from axioms,

[Virgil]

In article <a8244e3f-95e2-1312...@att.net>,

Jim Burns <james....@att.net> wrote:

> On 7/11/2016 2:09 PM, Virgil wrote:
> > In article <nm0n4n$crk$1...@news.albasani.net>,
> > Peter Percival <peterxp...@hotmail.com> wrote:
> >> Virgil wrote:
>
> >>> If something is an axiom then it is, by definition,
> >>> NOT a theorem,
> >>
> >> The sequence which consists of an axiom, and nothing else,
> >> is a proof of itself. So every axiom is a theorem.
> >
> > It is true because it is assumed, not because it has
> > been proved as following from something else,
> > which is my definition of theorem.
>
> Virgil: I hope that your long association with WM has not infected
> you with Mueckenheimlichkeit. Please be careful.
>
> Including the axioms as theorems is usually what we mean
> by theorems.

Wiki says otherwise

Theorem - Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Theorem?
In mathematics, a theorem is a statement that has been proven on the
basis of previously established statements, such as other theorems蟻nd
generally accepted statements, such as axioms. A theorem is a logical
consequence of the axioms.

[Peter Percival]

What I wish to claim is that "theorem" in the context of this thread (
:-) :-) ) means theorem in the sense of formal logic. And in that sense
every axiom is indeed a theorem. If you are using "theorem" in some...
ah hem... vaguer sense, then so be it.

[Dan Christensen]

On Monday, July 11, 2016 at 4:01:15 PM UTC-4, Jim Burns wrote:
> On 7/11/2016 2:41 PM, Dan Christensen wrote:

> > The important
> > thing here is that, for the purpose of formally developing
> > number theory, Peano's Axioms along with the ordinary rules
> > of logic and set theory have been shown to be quite sufficient
> > as a starting point. Each of these axioms is a truly
> > self-evident property of the natural numbers, and no others have
> > been found to be necessary. In that sense Peano's Axioms define
> > the set of natural numbers.
>
> Then you should accept
> ( { 1, 1/2, 1/4, ... }, 1, x |-> x/2 )
> as using the natural numbers. Do you?
>

No.

> > I don't find it particularly interesting that you can rename
> > all the numbers in some way and still get the same results
> > (with different names, of course). Maybe you can add that as
> > an axiom. I wouldn't bother.
>

Perhaps you didn't realize that if Peano(N,S,1) and Peano(N',S',1') (as defined above) then there exists a simple bijection f: N <--> N' such that

1 <--> 1'
S(1) <--> S'(1')
S(S(1)) <--> S'(S'(1))
...

Really just a simple renaming of the elements of one structure to obtain the other. Yes, it may be interesting at some level, but not very useful in number theory.

Bottom line, all you really need to formally develop number theory is Peano's Axioms (which define the set of natural numbers) and the ordinary rules of logic and set theory.

[Jim Burns]

On 7/11/2016 4:08 PM, Virgil wrote:
> In article <a8244e3f-95e2-1312...@att.net>,
> Jim Burns <james....@att.net> wrote:
>> On 7/11/2016 2:09 PM, Virgil wrote:
>>> In article <nm0n4n$crk$1...@news.albasani.net>,
>>> Peter Percival <peterxp...@hotmail.com> wrote:
>>>> Virgil wrote:

>>>>> If something is an axiom then it is, by definition,
>>>>> NOT a theorem,
>>>>
>>>> The sequence which consists of an axiom, and nothing else,
>>>> is a proof of itself. So every axiom is a theorem.
>>>
>>> It is true because it is assumed, not because it has
>>> been proved as following from something else,
>>> which is my definition of theorem.
>>
>> Virgil: I hope that your long association with WM has not infected
>> you with Mueckenheimlichkeit. Please be careful.
>>
>> Including the axioms as theorems is usually what we mean
>> by theorems.
>
> Wiki says otherwise

Actually, no, this doesn't say otherwise.
This doesn't say what a proof _is_ .

The point Peter Percival and I make is that writing down an axiom
by itself _is a proof_ , a perfectly fine formal proof of that axiom.

> Theorem - Wikipedia, the free encyclopedia
> https://en.wikipedia.org/wiki/Theorem?
> In mathematics, a theorem is a statement that has been proven on the

> basis of previously established statements, such as other theorems‹and

> generally accepted statements, such as axioms. A theorem is a logical
> consequence of the axioms.
>

So, what is "a statement that has been proven"?
<https://en.wikipedia.org/wiki/Formal_proof>
<quote wiki>
A formal proof or derivation is a finite sequence of
sentences (called well-formed formulas in the case of
a formal language) each of which is an axiom, an assumption,
or follows from the preceding sentences in the sequence by
a rule of inference. The last sentence in the sequence is
a theorem of a formal system.
</quote wiki>

Note that an axiom written down is "a finite sequence of sentences
([...]) each of which is an axiom, [...]." Thus, a proof.

It doesn't matter in any strong sense whether what _we_ call
"theorems" is what _you_ call "theorems and axioms" as long as
you and we understand each other, and as long as you read others
who do not share your notion while keeping in mind that they
mean something you do not.

It doesn't matter in any strong sense, but it still seems
inconvenient for you and liable to waste everyone's time
chasing down miscommunications that didn't need to be.