On 5/26/2012 9:35 AM, John Mikes wrote:
Brent wrote:
/1. Presumably those true things would not be 'real'. Only provable things would be true of reality./

Just to be clear, I didn't write 1. above.  But I did write 2. below.

/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'.

As Bruno said, "Provable is always relative to some axioms and rules of inference. It is quite independent of "true of reality". Which is why I'm highly suspicious of ideas like deriving all of reality from arithmetic, which we know only from axioms and inferences.

#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,

I see no reason to suppose that. Everything ever expressed so far has been done with a finite part of arithmetic. Assuming every integer has a successor is just a convenience for modeling things; you don't have to worry about running out of counters. There is a book "Ad Infinitum, The Ghost in Turing's Machine" by Rotman that proposes what he calls "non-euclidean arithmetic" which does not assume the integers are infinite. I can't really recommend the book because most of it is written in the style of French deconstructionist philosophy, but the Appendix has some interesting ideas.

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.

Axioms are just part of a logical, i.e. self-consistent, system. Mathematicians don't even care if they are "true of reality". They may or may not refer to imagined things; they are just assumed true for some inferences. I could take "I am typing on a keyboard" as an axiom, which I also happen to think is true, or I could take "I am a projection in a Hilbert space" which might be true, but is much more dubious.

In another logic than human, in another figment of a "physical world" different axioms would serve science.

Logic is about the relations of propositions, statements in language. Humans already have invented different logics.

2+2=4? not necessarily in the (fictitious) "octimality" of the '[Zarathustran' aliens in the Cohen-Stewart books
(still product of human minds).


"The world consists of 10 kinds of people.  Those who think in binary and those 
who don't.

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