FOM: new inductions unimportant

[Harvey Friedman]

Response to McLarty, 1:22PM 2/26/99:

You fail to make the usual distinction between:

1) ZFC does not prove "every finitely axiomatized fragment of ZFC is
consistent" and
2) ZFC proves the "consistency of each of its finitely axiomatized fragments."

Item 1) is what you meant. But why didn't you just say "ZFC does not prove
ZFC is consistent"? The fact that you didn't write "ZFC does not prove ZFC
is consistent" led me to think that you were confused.

Item 2 is what you denied, and it *normally* is parsed to mean:

2') for each finitely axiomatized fragment T of ZFC, ZFC proves T is
consistent.

And 2),2)' are true; in fact, they are provable in EFA.

Such distinctions and terminology are now more or less common, and you
might as well adhere to them.

>The step from individual fragments to the single assertion about "every
>finitely axiomatized fragment" requires induction that cannot be done within
>ZFC.

I fail to see the significance of this remark. It isn't the induction that
cannot be done within ZFC. It is that the language of ZFC does not express
the satisfaction relation for set theoretic statements. Obviously, you
cannot expect ZFC to prove an instance of induction that it cannot even
state!

>Nearly everyone on this list accepts that further induction. I
>certainly to. Only some radical finitists may question it in some way.
>Analoguous facts apply to any standard formal foundation for mathematics

But no one accepts this further induction until after they first introduce
set theories with unrestricted quantification over sets. In other words,
people only accept this further induction after they first do
metamathematical constructions and reflection.

>No consistent formal, first order, axiomatic theory includes every
>case of induction that you yourself will want to accept.

I think this is false and/or misleading. A system like ZFC may well include
far more than every case of induction that you yourself would accept as a
mathematician - perhaps not all that you would accept as a mathematician
engaging in metamathematical reflection.

This is a key reason behind the overwhelming power of the existing standard
formal systems such as ZFC. The fact that one can trivially cook up proper
extensions of ZFC routinely in no way, shape, or form reduces the crucial
importance of systems such as ZFC. Of course, there are also far reaching
extensions of ZFC via large cardinals - and that is again a major matter.