Re: ZFC

In article <haberg-0506051533320001@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hans Aberg <haberg@xxxxxxxxxx> wrote:
>Most of it, like the stuff discussed in this group, I gather. If you
>though go to math researchers in category theory, homological algebra and
>stuff, it is not so clear what to use. The interested one can check out
>the group sci.math.research, where the question pops up every now and then.

Even in these areas, there are standard dodges that allow most things to be
formalized in ZFC. Though it's true that there are some unclear issues at
the outer limits.

>>so ZFC is considered a "foundation for mathematics."
>
>This might be formally true if, in addition, the question of the
>consistency could be resolved.

ZFC is still considered a foundation for mathematics regardless.

It's true that if you insist to clinging to the intuition behind Hilbert's
program, maintaining that (1) a mathematical issue cannot be "resolved"
unless it is given a classical mathematical proof, and (2) the consistency
of a proposed foundation for mathematics must be "resolved" before that
foundation can be accepted, then you're still going to be left hanging,
probably forever. But most people have learned to let go of one or the
other of these intuitions and accept ZFC as a foundation for mathematics.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
.

Relevant Pages

  • Re: (Not quite) Cantors diagonal proof
    ... Josh Purinton wrote: ... > Not in ZFC. ... > mathematics, a mathematician could say to a crank, "don't bother me ... Lemma "ruclem39" refers to 38, ...
    (sci.math)
  • Re: (Not quite) Cantors diagonal proof
    ... Josh Purinton wrote: ... > Not in ZFC. ... > mathematics, a mathematician could say to a crank, "don't bother me ... Lemma "ruclem39" refers to 38, ...
    (sci.logic)
  • Re: An uncountable countable set
    ... Still, given your thoughtful posts, I am curious to know in which ways you think he is wrongheaded in his conception, if not simply by being over-technical when discussing general concepts. ... My disagreement with Virgil have to do with his comments about what mathematicians attempt to do with proofs, what is the significance of the existence of some formal derivation in this or that theory, what grounds do we have to think ZFC is consistent and so forth. ... Virgil's simplistic ideas are not peculiar to him, and have widespread currency among mathematicians who, as a rule, wish not to get tangled in a web of difficult conceptual and philosophical questions but simply want to get on with the mathematics. ...
    (sci.math)
  • Re: Countable models of ZFC
    ... I wonder which model of ZFC we're working in? ... It is the axioms that makes sense of "x e y". ... don't understand mathematics. ... I wouldn't be inclined to call the intended model a model, ...
    (sci.logic)
  • Re: what makes it true?
    ... I can elucidate the definition with an intuition, ... to be a curve. ... Since ZFC is usually used for mathematics, ... agree on some axioms for mathematics and get down to business. ...
    (sci.math)