Re: Frege: Reason's nearest kin


abo wrote:
One requires in addition appropriate definitions of 0, the successor
relationship, and the set of natural numbers. (I imagine this point is
made by Zalta outside the snip.)

I rebutted:
Definitions, by definition, are NOT additional principles.
They are usually just abbreviations. And the whole point
of the theorem is that given Hume's Principle as an axiom,
YOU CAN perform these definitions, in this framework, in
second-order logic.

What does perform mean?

This is a stupid question. If you know a definition when you see
one, then every time you have seen one, you saw whoever was
asserting/stating/defining it PERFORMING an act of definition.

Anyway, making or not making the definition is
independent of assuming or not assuming Hume's Principle.

Of course, but you can't do ANY of it without assuming Hume's
Principle. THERE, and NOT upon the definitions, is where the
content lies.

One can make the Frege-Arithmetic definitions of 0,
the successor relationship,

right

and the natural numbers

wrong

in second-order logic, augmented with a functional
symbol # which takes predicates as an argument, full stop

IDIOT, you do NOT get to full-stop THERE!
HOW DO YOU THINK *any* logical language EVER
gets augmented with any symbols?!?!?!? It gets augmented
BY YOUR ASSUMPTION OF AN *AXIOM* mentioning the symbol!

- Hume's Principle is not needed.

It IS SO TOO needed because THAT is where "#" GETS INTRODUCED!
UNTIL you "need" Hume's Principle, you DON'T HAVE "#" in your language!

If you try to do this without Hume's Principle then you have to make
"# a LOGICAL symbol, which completely begs the question of whether
numbers can be defined by logic.

Let's separate your previous statement where it actually NEEDS it:

in second-order logic, augmented with

We first have to stop HERE. As I said before, second-
order logic ALREADY COMES WITH alphabet and some formation
rules. You may NOT augment it with ANYthing.

a functional symbol

If you want to add new function symbols, then of course you may
do that, but you do NOT add them to the LOGIC! You MENTION
and USE them, and THEREBY INDIRECTLY DEFINE them, in NEW
AXIOMS! THAT is THE ONLY way in which new symbols can get added!

#

Since this is the ONLY non-logical symbol (with the possible exception
of equality, which is in fact also definable -- as indiscernibility)
being used
in this entire theory, it is actually more appropriate to take the
WHOLE theory
as one long DEFINITION of #. Of course, since the theory ONLY has ONE
non-logical axiom ANYway (unless you want to add Leibniz's Principle,
i.e., a definition of equality as indiscernibility) taking THAT axiom
as the
definition is even MORE appropriate. And it really is definitional to
be
specifying (precisely as HP does) the conditions under which 2
different
#'s (or applications of ANY function) must come up equal.

which takes predicates as an argument,

That is utterly redundant; in 2nd-order logic, 2nd-order predicates are
first-class citizens; this is merely acknowledging what the type of #
is,
something that would've been OBVIOUS FROM ITS OCCURRENCE in
the axiom (HP), IF ONLY YOU HAD HAD SENSE ENOUGH to realize
that you CAN'T just introduce symbols for the fun of it, and you HAVE
to introduce them BY using them in axioms.

full stop

Better late than never.

Note that in what I wrote, I did not claim these definitions were
additional axioms.

Of course you did. You claimed that they were
necessary. By definition, if they are definitions, they
are NOT necessary. You can say EVERYthing that you have
said WITH a defined term, WITHOUT that defined term.
You just SUBSTITUTE the DEFINITION of the defined term
FOR the defined term, and you get a sentence THAT MEANS THE SAME
THING! BY DEFINITION! *OF* "definition"!
Unfortunately, you personally are so stupid that you don't
know what a definition is.



0 in particular hardly needs a "definition".
It is just a constant. x=/=x is a "concept" that already
exists in the framework of 2nd-order-logic-with-equality,
completely IRrespective of whether anyone bothers to
baptize it by the name of the constant 0.

{x : x != x} is indeed part of the language of second-order logic.
However, unless you have access to a constant symbol 0 and an
appropriate definition of 0, you are not able to prove the Peano Axioms
which involve zero,

What UTTER bull***. OF COURSE you can prove them
You don't have to SPELL them with 0 for them to BE LOGICALLY
EQUIVALENT.

e.g. N0.

E.g. shut your ignorant yap.

Put baldly and most simply, you cannot
prove N0 unless you have appropriate definitions of "N" and "0".

This is almost true, but THE POINT is, GIVEN that you have those
definitions, you do NOT NEED THE NAMES "N" and "0"!
You can just prove things about the concepts directly, WITHOUT
abbreviating them! DEFINITIONS ARE FOR abbreviation!

Please note that I am more than happy to continue this conversation as
long as it continues to be a conversation. If I get shouted at or
called insults, I will stop replying.

Promises, promises......

.

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