Re: Scott and George's Teaching Thread

On Aug 22, 4:59 pm, Scott <ToaTe...@xxxxxxxxx> wrote:
I agree the paradigm is worthwhile learning; otherwise, I wouldn't be
here.

OK. Since you seem to be coming from an implementation-
of-programming-languages perspective, the first thing to do is
highlight the similarities between this and programming languages.
This paradigm is the one from which programming languages in general
were derived. Most programming languages that you have already
encountered have some sort of linguistic construct like a "boolean
expression".
First-order languages include those in things like (P ^ Q), (P v Q),
~P, ~Q,
(P --> Q), etc. Actually, this part is 0th-order but the point is, 1st
includes all
of 0th, and the above are also 1st-order formation rules in that if P
is already
a 1st-order wff, then (P ^ Q) is as well. "wff" abbreviates "well-
formed formula".

In a construct like f(x,y,z), this whole thing is a TERM that is
intended to denote some element of the DOMAIN.
This is a term whenever f is ternary function-symbol and x,y, and z
are terms.
So I probably should've written it f(t1,t2,t3).

Terms denote elements of the domain and PREDICATE-symbols are, like
function
symbols, applied to a list of terms, but instead of resulting in a
bigger term, result
in an atomic sentence with a TRUTH-value.


The realm we are about to investigate, set theory,
ONLY HAS ONE predicate. Because it is binary, it is usually written
infix.
In other words, instead of writig I(a,b) to mean a IN b, (that would
be "capital I" for IN),
we write aeb. In Real Math Books, the set theoretic membership
predicate
is written as an infix italic lower-case Greek epsilon.
"e" is just my ascii approximation of that.

The definition of a first-order language gives rules for combining the
symbols to
ensure that the resulting terms and formulas are well-formed, but it
does not say
anything about WHAT symbols get to be so combined. The reason for
that omission
is simply that you are free to name your symbols basically anything
you like;
what anything IS CALLED simply has no effect on whether it is or isn't
well-
formed. Here, however, because we don't have upside-down or backwards
letters,
or greek letters, we have to take a little care.

The purpose of this enterprise is to prove theorems from axioms.
We don't prove the axioms. We just prove that IF you assume the
axioms, then there are certain other things (the theorems) that you
MUST
accept ALONG WITH them, that follow, inseparably attached, in their
train.
The symbols that you throw into the first-order-language-making-
machine
that was outlined on p.24 are the symbols (predicates and functions)
that occurred
in your STATEMENT OF YOUR AXIOMS. YOU HAVE TO STATE SOME AXIOMS
to get this process off the ground. There is a sense in which 0th-
order tautologies
and 1st-order validities do NOT depend on any particular set of
symbols, but the point
is, we're trying to be ABOUT something here, so it usually IS good to
restrict the
set of symbols to a known vocabulary in advance.

The cluster of symbols mentioned in your axioms is called the
SIGNATURE of your
1st-order language. A signature is what you need "as input to the
process" from p.24,
to produce A PARTICULAR 1st-order language as output.

Since you will be seeing = all the time, another important thing to
remember is that =
IS NOT a predicate in this language. There are axioms governing
equality, but since
we're doing set theory, we don't have those axioms, so we don't have =
in our signature.
When you see "=" in set theory, you need to think of it as a macro.
B=C
MEANS (when C and B are sets)
Ax[ xeB <--> xeC ].
It means "B is a subset of C and C is a subset of B".
The "=" is an abbreviation for something phrased PURELY in terms of e
(since,
this being set theory, EVERYthing is phrased in terms of set
membership).

There is another way of treating "=" that involves adding it to the
underlying
language as a logical symbol (it's STILL not a predicate, if you do
that); that
results in something called "first-order logic With equality". We
are not doing
that because we are using set theory, and set theory is powerful
enough to define
equality; you just saw the definition above.




.

Relevant Pages

  • Re: Scott and Georges Teaching Thread
    ... Please think of this as a programming language. ... predicate in this language is e. ... The axioms jointly CLARIFY AND EXPLAIN ... Some of these logical primitives are from 0th-order logic; ...
    (sci.logic)
  • Re: Small set Theory:final version.
    ... "P_defined" in the axioms to ... object language but only in meta language. ... predicate is - that there are at most countably many zuhair sets). ...
    (sci.math)
  • Re: Small set Theory:final version.
    ... Hi hagman, I already modefied a lot of the theory that all of this ... "P_defined" in the axioms to ... object language but only in meta language. ... predicate is - that there are at most countably many zuhair sets). ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... > first-order language of set theory is independent of ZF if it can't be ... >> that real is in the set of reals in ZF. ... axioms schemas, I don't know where it all starts. ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... to consist mainly of bashing set theory "curing" Tarski's curse and what ... predicate for e.g. the language of arithmetic in that language ... predicate is so important to him: it's supposed to show that we can have ... a switch would look like. ...
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