Brent wrote:

*1. Presumably those true things would not be 'real'.  Only provable things
would be true of reality.*
*2. Does arithmetic have 'finite information content'?  Is the axiom of
succession just one or is it a schema of infinitely many axioms?*
Appreciable, even in layman's logic.

In '#1' -  I question "provable" since in my agnosticism an 'evidence' is
partial only, leaving open lots of (so far?) unknown/able aspects to be
covered. In the infinity(?) of the "world" also the contrary of an evidence
may be 'true'.

#2 is a technically precise formulation of what I tried to express in my
post to Bruno.
IFF!!! "anything"  (i.e. everything) can be expressed by numerals, the
information included into arithmetic * IS* infinite, however as it seems:
in our (restricted) view of "the world" (Nature?) there seem to be NO
numbers to begin with.
In our human 'translation' we see 1,2, or 145, or a million "OF SOMETHING"
- no the (integer?) numerals.

Axioms? in my vocabulary: imagined things, necessary for certain theories
we cannot substantiate otherwise.
In another logic than human, in another figment of a "physical world"
different axioms would serve science.
2+2=4? not necessarily in the (fictitious) "octimality" of the
'[Zarathustran' aliens in the Cohen-Stewart books
(still product of human minds).

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