[FOM] First-order arithmetical truth

Arnon Avron aa at tau.ac.il
Sun Oct 22 18:22:14 EDT 2006

Dear Vladimir,

On Sat, Oct 21, 2006 at 09:42:07PM +0100, V.Sazonov at csc.liv.ac.uk wrote:
> Dear Arnon,
> In fact I asked about clear criteria how to check whether anybody has 
> this ability. 

Please give me a clear criteria for being a "clear criteria",
and if your criteria are really clear,
I'll  happy to try to give you what you ask for.

> This could be a precise definition of the 
> standard model, 

Please give me clear criteria for being a  "precise definition",
(this might be a precise definition of a "precise definition").

> if possible at all. But I see that you rather present your subjective 
> beliefs and appeals to beliefs of others.

Please give me a non-subjective definition of (or 
a clear criteria for)  "subjective". 

> Does "almost every person" know and understand what is 
> induction axiom 

Definitely. S/he  might not know the name "induction",
and might not know that s/he is using the logical rule of induction
(logical in the language with transitive closure operation) -
but nevertheless s/he basically  understands and knows the rule/axiom.
Thus almost every person understandis without being told, 
that   had wisdom been a property that
a male child should necessary have whenever his father does,
then all the male descendants of a wise man should necessarily
be wise (if you dont believe me that everyone understands
this  - try it out). Similarly,
you can explain to almost every person the so-called heap paradox,
by just giving him/her the two premises ("every heap consisting of one
grain of sand is small" and "by adding exactly one grain 
of sand to a smal heap we get another small heap") and pointing
out the obvious conclusion ("every heap of grains of sand
is small"). Again almost every person would immediately 
see that the obviously false conclusion does indeed follow 
from the seemingly true premises  (in my opinion:
*logically* follows from them), even though not every person
would realize that s/he is using mathematical induction 
for this simple inference.

By the way, I think that your notion of a "feasible" number
is meaningless because it leads to a *real* paradox
of the "heap" type. Once you give a precise definition
of "a feasible number", or provide  clear, objective
criteria for being a "feasible" number, I'll be able
to prove that every natural number is "feasible".

>  Even the majority of professionals have 
> no idea on  nonstandard numbers and understand induction axiom 
> not in the same way 
> as logicians, knowing nothing about subtle distinctions. Definitely, 
> all of us have somewhat different pictures and intuitions on N in our 
> minds.

Simply not true. 

> That is, the naive 
> concept of N and mathematical one are quite different.

They are identical (at least from a certain age onwards). 
Once a child realizes that the process of counting potentially 
has no end  (and every nornmal child realizes it with astonishhment
at a certain point) s/he has the mathematical  concept of N. 

To the rest of your message I shall not respond, because our
views are so different, that the only things I can say again and again
are of the sort I said above: "I dont agree" or "not true". This
would not be very productive.


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