Repeating myself on RVM

[martdowd at aol.com]

Haim Gaifman wrote:
 

But the notion of the absolute totality of all subsets of N, 
...,  is highly suspicious.
 
 It did not seem so to Cantor, indeed formalizing the notion was an incentive for early set theory.
Martin Dowd
-----Original Message-----
From: Haim Gaifman <hg17 at columbia.edu>
To: fom at cs.nyu.edu
Sent: Mon, Aug 3, 2020 4:41 pm
Subject: Re: Repeating myself on RVM

 Here are some points concerning the question posed by Joe Shipman:
  (1) The very notion of "being too restrictive" is misleading. At best, the argument in which it is used is question begging.
 The foundational debate about CH is not about the true cardinality of P(N)  (where P(N) = power set of N, and N=set of natural numbers)
 but about the very meaningfulness of the notion of a set that contains all subsets of N.
 
 We know  of subsets of N because we can construct them using particular definitions. We can also define families of subsets, by using
 definitions the depend on parameters ranging over N. But the notion of the absolute totality of all subsets of N, 
 subsets whatsoever,  irrespective of a way of getting them by any definition or construction,  is highly suspicious.
 If nonetheless you accept this notion, then of course, V=L might appear restrictive. But the debate is about
 this very legitimacy of the very notion of P(N).
 
 (2) V=L legitimizes the notion of an arbitrary set, if you assume an absolute notion of well-ordering. (3) It appears, by now, that attempts to answer the question about the cardinality of P(N), by assuming some additional set-theoretic axiom,
 are doomed to fail, because  any such  axiom  will not be "obvious". (4) Why should we accept the axiom about extending Lebesgue measures, as a set theoretic axiom? Haim Gaifman
  On 8/2/2020 11:59 PM, Joe Shipman wrote:
  
 I read a lot of papers which talk about the unsatisfactoriness of “new axioms” for non-absolute statements like CH. It seems clear that most people working in the area don’t like V=L and related axioms because they are too restrictive about that sets may exist, and feel like prospects for settling CH are dim.

But I have a still never heard a satisfactory explanation of what is wrong with the axiom that Lebesgue measure can be extended to all sets of reals in a way that remains countably additive (though no longer translation-invariant).

What is an example of an independent-of-ZFC statement anyone cares about that this axiom does NOT decide (apart from propositions implying the consistency of cardinals larger than “measurable“)?

— JS

Sent from my iPhone
 
 
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