[FOM] ACA0, PA, PA'

[Dean Buckner]

Harvey,

Thanks for the note.  I understand perfectly what you're saying, but you
don't understand perfectly what I'm saying.

Of course: if we need to define the term "natural number", we are going to
to run into the sort of problem I am complaining about.  We can perfectly
well say "a natural number is anything that is obtained from 0 by the
repeated addition of 1".  But the word "repeated" cannot itself be defined.
It represents a rule which has to be learned as part of learning a (natural)
language.  It doesn't represent some abstract indefinite extension of "1,
1+1, 1+1+1".  Not at all, its meaning is embodied in the sign "1, 1+1, 1+1+1
...", whose meaning is perfectly exact, but governed by different rules from
the rules governing "1, 1+1, 1+1+1", as I argued some postings ago (and as
Wittgenstein argued about 70 years ago).

Earlier I supplied a proof that

    p | (a.1 & a.2 ... & a.n) implies p | x, where x is oneof (a.1 & a.2 ...
&
a.n)

from     "p | (a & b) implies p | a  or  p | b" where "x is oneof (a1 & a2
... & an)" is defined as "x=a.1 or x=a.2 ... or x=a.n".

We infer from "p | (a.1 & a.2 ... & a.n)" that

    p|a.1 or p|a.2 ... or p|a.n

using the rules for plural quantification, whereby operations on "compound"
proper names are understood to be shorthand for propositions involving
simple proper names.  Then we infer

    p|x where x=a.1 or x=a.2 ... or x=a.n

then (from the definition of oneof)

    p|x where x is oneof (a.1 & a.2 ... & a.n)

Nowhere in this proof do we require a definition of number, or a rule of
induction or anything of that sort.  We grasp the proof simply by
understanding how to use the terms which embody the induction principle.
(We learn their grammar).  Plural expressions have a rule of induction built
into them.  They are inherently finite. And so we cannot define their
meaning as we define the meaning of "unicorn".

Karam sipo tatura.


Dean




Dean Buckner
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ENGLAND

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