[FOM] Improving set theory

[Joe Shipman]

For the purposes of ordinary mathematics, though, quantification over the totality of all sets is almost never necessary. You can either assume a very small large cardinals axiom (an inaccessible limit of inaccessibles satisfies the Grothendieck Universes Axiom and practically anything mathematicians care about is absolute enough that that will suffice), or you can work in V_alpha for some large but accessible alpha.

If you still have trouble with meaningfulness, then forget about Replacement and let your “alternative” be ZC. That’s still got the same advantages over other foundational schemes, unless you can tell us of a theorem that ZFC needs Replacement for  but which an alternative foundational scheme handles “meaningfully”.

— JS

Sent from my iPhone

> On Jan 11, 2020, at 12:39 AM, Paul Blain Levy <P.B.Levy at cs.bham.ac.uk> wrote:
> 
> Dear Harvey,
>> Date: Thu, 9 Jan 2020 13:10:02 -0500
>> From: Harvey Friedman <hmflogic at gmail.com>
>> To: Foundations of Mathematics <fom at cs.nyu.edu>
>> Subject: [FOM] Improving Set Theory
>> Message-ID:
>> <CACWi-GVffjWwi_-KLO5JDiQBDMKHcKAFUbgPAJUpJ4mKU0pM6A at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> 
>> ZFC has become the standard foundation for mathematics since about
>> 1920. Alternatives have been proposed but not widely endorsed at least
>> not yet.
> You are right.  We critics of ZFC have work to do.
>> I am particularly interested in what people think is lacking or is
>> flawed about ZFC.
> Nothing is lacking.  ZFC is not too weak but too strong.  The problem is that its language allows quantification over the entire totality of sets.  In my view (and others have said the same), that totality does not exist, so the language is not really meaningful.
> 
> Paul
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