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Re: What's formal, axiomatic mathematics without the word "infinity"?

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On Tuesday, 20 June 2023 at 10:10:04 UTC+2, Mikko wrote:
> On 2023-06-19 05:50:57 +0000, Khong Dong said: 
> 
> > https://qr.ae/pyPSmu 
> 
> Ordinary formal first order logic does not define "infinity",

Indeed, it's strictly and rigorously finitary: terms (statements)
must be finite, proofs must be finite, the very notion of recursivity
is all about what can be computed with strictly finitary means
only, i.e. what is really doable.

> and the 
> usual meaning of "infinity" is not definable there.

As for semantics, so building a "theory" on top of that
"language", *every* "meaning" is potentially definable:
whether one makes an interesting theory out of it or not.

(A true finitist would object that infinite objects do not
exist at all, not even ideally/abstractly/in principle.  I
won't comment on that, just to say that that's still a
complain about the "theory", the underlying "logic" is
finitary for everybody and not in question.)

(Those who complain about "logic", rather complain
about the non-constructivity of classical logic: but that
is still an issue of "theory" and not of the underlying
"language": though this is not a trivial point.)

> In particular, the first order Peano arithmetic does not
> define "infinity".

In particular, that's obvious nonsense, given that PA has
an induction axiom so that << [a] model of the Peano
axioms is a triple (N, 0, S), where N is a (necessarily infinite)
set >> (from WP), which then becomes the *Axiom of
Infinity* in a set theory.

TL;DR  (Formal, mathematical) "logic" is necessarily finitary,
Logic or Mathematics or what have you is just another level
and needn't be.

Julio

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Thread

What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-18 22:50 -0700
  Re: What's formal, axiomatic mathematics without the word "infinity"? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-19 06:58 -0700
    Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-19 11:06 -0700
      Re: What's formal, axiomatic mathematics without the word "infinity"? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-19 19:02 -0700
  Re: What's formal, axiomatic mathematics without the word "infinity"? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-19 08:01 -0700
    Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-19 17:24 -0700
      Re: What's formal, axiomatic mathematics without the word "infinity"? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-20 11:37 -0700
        Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-20 11:57 -0700
          Re: What's formal, axiomatic mathematics without the word "infinity"? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-20 12:06 -0700
            Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-20 12:12 -0700
              Re: What's formal, axiomatic mathematics without the word "infinity"? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-20 13:39 -0700
                Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-20 15:33 -0700
  Re: What's formal, axiomatic mathematics without the word "infinity"? Mikko <mikko.levanto@iki.fi> - 2023-06-20 11:10 +0300
    Re: What's formal, axiomatic mathematics without the word "infinity"? Julio Di Egidio <julio@diegidio.name> - 2023-06-20 03:20 -0700
  Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-20 10:19 -0700
    Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-20 10:42 -0700
    Re: What's formal, axiomatic mathematics without the word "infinity"? Julio Di Egidio <julio@diegidio.name> - 2023-06-20 16:07 -0700
      Re: What's formal, axiomatic mathematics without the word "infinity"? Khong Dong <khongdongphong@gmail.com> - 2023-06-21 08:12 -0700
        Re: What's formal, axiomatic mathematics without the word "infinity"? Julio Di Egidio <julio@diegidio.name> - 2023-06-22 03:08 -0700

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