903: Foundations of Large Cardinals/1

[Harvey Friedman]

Here we develop a kind of foundations of large cardinals. We are able
to generate powerful large cardinals just through ELEMENTARY
EQUIVALENCE. This avoids the use of elementary extensions.  Elementary
extensions are certainly useful for philosophy stories, but they make
certain philosophical stories more cumbersome and less immediately
compelling.

DEFINITION 1. A relation is a binary relation R, where we write x R y.
fld(S) is {x: (therexists y)(x R y or y R x). A  completion of R is a
relation S containing R, where every subset of fld(R) is the set of S
predecessors of some element of fld(S).

DEFINITION 2. A completion tower is a nonempty linearly ordered set of
relations, where if R < S then S is a completion of R.

THEOREM 1. No relation is a completion of itself. If S is a completion
of R then 2^|R| <= |S|. Every completion tower is well ordered.

We now wish to associate a natural two sorted relational structure to
any completion tower T = (X,<).

DEFINITION 3. Let T = (X,<) be a completion tower. T* is the two
sorted structure (X,<,E,app), with sorts X, and E = the union of the
fields of the R in X. < is from T, and app is the ternary relation
between X and E and E, given by app(R,x,y) iff x R y. We say that two
completion towers T_1,T_2 are first order equivaelnt if and only if
T_1* and T_2* are elementarily equivalent as two sorted structures.

We now investigate first order equivalence between various T*. Let us
start with the simplest nontrivial situation which is with a
completion tower T of cardinality 2. We ask that the two one element
restrictions of T be first order equivalent. This already packs some
logical power.

PROPOSITION 1. There is a completion tower of cardinality 2 whose one
element restrictions are first order equivaelnt.

THEOREM 2. Proposition 1 is provable in KP + "for all ordinals alpha <
(2^omega)+, V(alpha) exists".

CONJECTURE. Proposition 1 over KP is mutually interpretable with
roughly KP + "for all ordinals alpha of cardinality at most c,
V(alpha) exists".

PROPOSITION 3. Let k be given. There is a completion tower of
cardinality k whose equal cardinality restrictions are first order
equivalent.

THEOREM 4. Proposition 3 is provable in KP + "for all ordinals alpha <
beth_omega. V(alpha) exists".

CONJECTURE. Proposition 3 over KP is mutually interpretable with
roughly KP + "for all ordinals alpha < beth_omega. V(alpha) exists".
Or maybe a slight variant.

PROPOSITION 5. Let k be given. There is a completion tower of order
type omega whose cardinality k restrictions are first order
equivalent.

THEOREM 6. Proposition 5 is provable in KP + "for all ordinals alpha <
beth_omega. V(alpha) exists".

CONJECTURE. Proposition 5 over KP is mutually interpretable with
roughly KP + "for all ordinals alpha < beth_omega. V(alpha) exists".
Or maybe a slight variant.

PROPOSITION 7. There is a completion tower of order type omega whose
equal sized finite restrictions are first order equivalent.

THEOREM 8. Proposition 7 is provable in ZFC + there exists kappa such
that kappa arrows omega.

CONJECTURE.  Proposition 5 over KP is mutually interpretable with
roughly ZFC + there exists kappa such that kappa arrows omega.

PROPOSITION 8. There is a completion tower of order type omega whose
equal sized restrictions are first order equivalent.

THEOREM 10. ZFC + Proposition 8 is interpretable in ZFC + there exists
a measurable cardinal.

CONJECTURE. Proposition 8 is provable in some standard large cardinal
hypothesis.

PROPOSITION 11. There is a completion tower of order type omega + 1
whose order type omega + 1 restrictions are first order equivalent.

THEOREM 12. Proposition 11 is interpretable in ZFC + some strongly
inaccessible cardinal is greater than some measurable cardinal.

CONJECTURE. Proposition 11 is mutually interpretable with roughly ZFC
+ there exists a measurable cardinal.

PROPOSITION 13. There is a completion tower of order type omega + 1
whose equal type restrictions are first order equivalent.

CONJECTURE. Proposition 12 is mutually interpretable with roughly ZFC
+ there exists many measurable cardinals.

DEFINITION 4. Let T be a well ordered completion tower. A regular
restriction is a restriction of T whose proper initial segments are
finite unions of intervals.

PROPOSITION 14. Let alpha be given. There is a completion tower of
order type alpha whose pairs of regular restrictions of the same order
type are first order equivalent.

CONJECTURE. Proposition 14 is mutually interpretable with roughly ZFC
+ there are measurable cardinals of familiar strong kinds.

*MAKING REVERSALS EASIER*

We can make these conjectured reversals easier by strengthening the
notion of completion. Here are some additional conditions.

1. Require extensionality for the relations.
2. Require elementary extension.

With 2, even Proposition 1 would need roughly a strongly inaccessible,
and not be provable in ZFC.

But I prefer not to use elementary extensions anywhere, just
elementary equivalence.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 903rd in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-899 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

900: Ultra Convergence/2  10/3/21 12:35AM
901: Remarks on Reverse Mathematics/6  10/4/21 5:55AM
902: Mathematical L and OD/RM  10/7/21  5:13AM

Harvey Friedman