Higher Set Theory – Cantorian Sets / Large Cardinals in the Infinite

[heff001]

Zermelo-Fraenkel Axioms - the Axiom of Choice (ZFC), is conceptually incoherent. To me, they stole Cantor’s brilliant work and minimized it. Replies?

 

[mathman]

You might want to expand on your point. It is completely opaque as stated.

 

[heff001]

>Cantor's work and Platonism
Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do 'numbers and sets'.

>ZFC is a refutation of Platonist set theory in general
Set-theoretic intuition, as formalized in the Zermelo-Fraenkel axioms with the axiom of choice (ZFC), is conceptually incoherent.

Ref.
Truth, Proof and Infinity pp 13-23 | Peter Fletcher

Peter Fletcher lists the objections to the use of each below as a foundation for ZFC:

(1) sets as consistent multiplicities or multiplicities considered as unities
(2) sets as collections
(3) sets as classes, in the sense of extensionalized properties
(4) the limitation of size view
(5) the iterative conception of sets
(6) sets as an extrapolation from finite sets of physical objects
(7) sets obtained by a transition from potential to actual infinityIs Fletcher correct in doing so?

I am a student of Cantor right now and studying all he did between the nervous breakdowns...
I am totally puzzled by ZFC wording, purpose, the notion that Cantor's work can simply be captured in ZFC.

 

[heff001]

Any reply?

 

[PeterDonis]

It looks to me like you are asking about something that's a matter of opinion, not mathematical proof. As far as mathematical proof goes, we already know that ZFC cannot be proven to be consistent, nor can it be proven to be inconsistent. Whether ZFC captures your "intuitions about sets" is not a matter of mathematical proof, since "intuitions" can't be formalized--if they could be, they wouldn't be intuitions.

 

[heff001]

Perfect. Brilliant. Thank You.