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

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Discussion Overview

The discussion centers around the conceptual coherence of the Zermelo-Fraenkel Axioms with the Axiom of Choice (ZFC) in relation to Cantorian set theory and large cardinals. Participants explore philosophical implications, the nature of mathematical objects, and the limitations of ZFC in capturing Cantor's work.

Discussion Character

  • Debate/contested, Conceptual clarification, Meta-discussion

Main Points Raised

  • One participant argues that ZFC is conceptually incoherent and minimizes Cantor's contributions to set theory.
  • Another participant requests clarification on the initial claim, finding it opaque.
  • A participant introduces the idea of Platonism in mathematics, suggesting that ZFC contradicts the existence of abstract mathematical objects independent of human thought.
  • Peter Fletcher's objections to ZFC are listed, questioning its foundations and whether it adequately represents Cantor's work.
  • Concerns are raised about the inability to prove the consistency or inconsistency of ZFC, suggesting that personal intuitions about sets cannot be formalized within its framework.
  • A later reply expresses agreement with the critique of ZFC, indicating appreciation for the insights shared.

Areas of Agreement / Disagreement

Participants express differing views on the coherence of ZFC and its ability to encapsulate Cantorian set theory. There is no consensus on whether ZFC is a suitable foundation for set theory or if it undermines Cantor's contributions.

Contextual Notes

Participants highlight limitations in ZFC's ability to formalize intuitions about sets and question its foundational assumptions. The discussion remains open-ended regarding the implications of these limitations.

heff001
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Zermelo-Fraenkel Axioms - the Axiom of Choice (ZFC), is conceptually incoherent. To me, they stole Cantor’s brilliant work and minimized it. Replies?
 
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You might want to expand on your point. It is completely opaque as stated.
 
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>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.
 
Any reply?
 
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.
 
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Perfect. Brilliant. Thank You.
 

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