[FOM] Finite to Infinite

[joeshipman at aol.com]

In his numbered posting 808 from May 1, 2018, Harvey wrote:


"One thing I have worked on, announced in various talks, discussed on
the FOM, but did not write up proofs, is that, in a certain hard nosed
and perhaps surprising sense,


ZFC RESULTS FROM TAKING ALL TRUE STATEMENTS ABOUT THE HEREDITARILY
FINITE SETS OF A CERTAIN NATURAL FORM AND ADDING THE AXIOM OF
INFINITY.


I.e., ZFC naturally arises from going from the finite to the infinite.


This result(s) go back decades, but I want to revisit this with a
fresh (older) mind."




This inspired me to think about the issue more systematically, which led to the following questions that I hope Harvey can shed some light on.




If A is a recursively enumerable set of axioms in the language of set theory with equality, let us say that A is “adequate” if 


(1) V(omega) |= A  (the axioms are true about the Hereditarily Finite Sets)
(2) (A U AxInf) |- ZFC  (the axioms are true about V and with AxInf allow any ZFC axiom to be derived)


where AxInf is the standard ZFC version stating that an inductive set exists. 


An obvious candidate for A is simply ZFC\AxInf. This has the logical strength of PA (though one must add ~AxInf to get something bi-interpretable with PA). 


A “maximal” candidate for A could be 
A_max: all the statements Phi such that 
ZFC |- (Phi & (V(omega) |= Phi))  (we can prove Phi is true about the HF sets and about all sets)


This is much stronger than PA, but in a trivial way.


To go beyond this, one would need a sentence Psi such that either
i ) ZFC |- (V(Omega) |= Psi) and we haven’t proved either Psi or ~Psi yet
or
ii ) ZFC |- Psi and whether V(omega) |= Psi is unknown.


I don’t know a great example of either of these, although it is easy to make silly examples like "AxInf -> CH" or "either AxInf or a huge cardinal is consistent". To avoid silly examples, some syntactic restriction on Psi would make sense.


However, we would really prefer to have natural “maximally coherent” sets of sentences for our axiomatizations, not just single sentences.


Instead of asking for a maximal adequate A, we can also ask if there is any “natural” adequate A which is stronger than ZFC\Inf. If there are nicely axiomatixable theories of the HF sets corresponding in strength to systems above PA like the arithmetic parts of ATR_0 or Pi_1^1-CA_0, what are their axiomatizations? This seems unlikely to get us beyond ZFC though, when we add an axiom of infinity.


There are also obviously true statements about finite sets which we know are independent of ZFC: "for any set S with at least two elements, there is a subset of P(S) not in 1-1 correspondence with S or P(S)". This isn't too helpful, though, because it's probably more complicated than anything we can prove a metatheorem about; counting this as evidence against CH seems ad hoc.


In the other direction, we could ask for a minimal adequate A or at least for a natural adequate A strictly weaker than ZFC\AxInf, if that is possible. Can you get ZFC by adding AxInf to a theory of HF sets which corresponds in strength to EFA or RCA_0 or WKL?


Some goals:


1) put ZFC on a firmer foundation by showing that it comes it comes from AxInf + “something simpler than ZFC\AxInf”


2) put ZFC on a firmer foundation by showing that it comes it comes from AxInf + “something weaker than ZFC\AxInf” [not necessarily the same as 1!]


3) Find a natural (not contrived like some examples above) sentence or coherent collection of sentences true when relativized to V(omega) but undecided by ZFC (possibly a candidate for a new axiom or scheme extending ZFC if it is particularly natural or maximally coherent in some sense)


4) find a naturally axiomatizable subsystem of A_max stronger than ZFC\AxInf.


-- JS
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