Re: Does a potential infinity actually exist?

On Aug 23, 9:39 am, aatu.koskensi...@xxxxxxxxx wrote:
MoeBlee wrote:
If you would either give 'potential infinity' as a primitive and
axioms with it or 'potential infinity' defined from primitives, then
there might be something of specific mathematical interest there.

You'll find a systematic treatment of "potential infinity" in
intuitionistic mathematics, especially the theory of choice sequences.
Of course, as with any piece of mathematics, the principles that
concern choice sequences must be seen justified on the "informal"
intuitionistic understanding if we are to adopt them as axioms (of
intuitionistic analysis, say).

Intuitionistic analysis and intuitionistic mathematics in general will
probably not be to Petry's liking, though, highly abstract and
infinitary as they are.

Here's where I'm somewhat in the fog: As far as formal theories go, I
know that Heyting arithmetic is the intuitionistic answer to PA, but
that doesn't give us an intuitionistic analysis. So, okay, I know that
there are various proposed constructivist set theories (I think
specifically intuitionistic ones too?) (e.g., in Troelstra & van
Dalen), but I don't know where there is a treatment that derives
analysis from such a set theory (hmm, I seem to have recalled looking
at Kleene & Vesley's 'The Foundations Of Intuitionistic Analysis' and
not finding a formal axiomatization of analysis, though maybe I
misremember that). And though I am familiar with the notion of a
choice sequence and with various intutionistic ideas about analysis
(mainly, I've read, but not throroughly studied, Heyting's book
'Intuitionisim'), I've not seen a book that clearly derives
intuitionisitc analsysis from axioms, while, on the other hand, an
axiomatic set theoretic basis for classical analysis is found in
numerous textbooks.

I do know the intuitionistic predicate calculus, so I'd like to see
some exact set of axioms (whether set theoretic or otherwise) then to
derive intutitionistic analysis. And then in that language and with
those axioms, if there is a formal definition of 'potential infinty'
then I'd like to see that too. Also, as I understand, Bishop's school
is constructivist but not necessarily intuitionistic, and I don't find
an axiomatization in Bishop & Bridges nor in Beeson (maybe this school
even eschews axiomatization?) (I haven't seen Bridges & Richman). Then
there's Martin-Lof, who I just don't understand from the beginning; I
think I need more study to catch up to approaching his work.

MoeBlee


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Relevant Pages

  • Re: Does a potential infinity actually exist?
    ... axioms with it or 'potential infinity' defined from primitives, ... intuitionistic mathematics, especially the theory of choice sequences. ... Intuitionistic analysis and intuitionistic mathematics in general will ...
    (sci.math)
  • Re: Does a potential infinity actually exist?
    ... axioms with it or 'potential infinity' defined from primitives, ... There is a school in mathematics, called constructivism, ... So do you think the original poster's "potential infinity" ...
    (sci.math)
  • Re: Does a potential infinity actually exist?
    ... axioms with it or 'potential infinity' defined from primitives, ... There is a school in mathematics, called constructivism, ...
    (sci.math)