[FOM] Improving set theory

[Paul Blain Levy]

Dear Joe,

Consider the following ZFC theorems:

(1) Borel determinacy

(2) The set V_{omega_1} is well-orderable.

These are not theorems of ZC.  (Harvey proved that for Borel determinacy.)

But they are theorems of TOCS, a new system* that I advocate instead of 
ZFC.  See here:

https://arxiv.org/abs/1905.02718

No inaccessible cardinal is required to prove them.  TOCS is just a 
subsystem of ZFC (or can be seen as such).

To repeat, I do not criticize the axioms of ZFC (such as Replacement).  
It's the *language* of ZFC that is problematic, because of the 
unrestricted quantification.

Paul

* I am changing the name from TOPS to TOCS, which stands for "Theory of 
constructed sets".


> Date: Sat, 11 Jan 2020 00:56:54 -0500
> From: Joe Shipman <joeshipman at aol.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] Improving set theory
> Message-ID: <946711ED-144F-46B2-9B26-28B830F0C3B5 at aol.com>
> Content-Type: text/plain; charset=utf-8
>
> 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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