Re: Questioning the defintions of set and element.

In article <oClSj.1209$7z4.15@xxxxxxxxxxxxx>, Mark <user@xxxxxxxx> wrote:

I don't see how a logical theory can be based on the undefined.

Quoted from "Introduction to Logic and to the Methodology of Deductive
Sciences", by Alfred Tarski, translated by Olaf Helmer; Dover
Publications, Inc., New York, unabridged republication of the 9th
printing, 1961, of the 1946 second revised edition. ISBN
0-486-28462-X, pp 117 ff. What appears in the original in smallcaps
font I place in quotation marks.

We shall now attempt an exposition of the fundamental principles
which are to be applied in the construction of logic and
mathematics. The detailed analysis and critical evaluation of
these principles are tasks of a special discipline, called the
"methodology of deductive sciences" or the "methodology of
mathematics." For anyone who intends to study or advance some
science it is undoubtedly important to be conscious of the method
which is employed in the construction of that science; and we
shall see that, in the case of mathematics, the knowledge of that
method is of particular far-reaching importance, for lacking such
knowledge it is impossible to comprehend the nature of
mathematics.

The principles with which we shall get acquainted serve the
purpose of securing for the knowledge acquired in logic and
mathematics the highest possible degree of clarity and
certainty. From this point of view a method of procedure would be
ideal, if it permitted us to explain the meaning of every
expression occurring in this science and to justify each of its
assertions. It is easy to see that this ideal can never be
realized. In fact, when one tries to explain the meaning of an
expression, one uses, of necessity, other expressions; and in
order to explain, in turn, the meaning of these expressions,
without entering into a vicious circle, one has to resort to
further expression again, and so on. We thus have the beggining of
a process which can never be brought to an end, a process which,
figuratively speaking, may be characterized as an "infinite
regree" - a regressus in infinitum. The situation is quite
analogous as far as the justification of the asserted statements
of the science is concerned; for, in order to establish the
validity of a statement, it i snecessary to refer back to other
statements, and (if no vicious circle is to occur) this leads
again to an infinite regress.

By way of a compromise between that unattainable ideal and the
realizable possibilities, certain principles concerning the
construction of mathematical disciplines have emerged that may be
described as follows.

When we set out to construct a given discipline, we distinguish,
first of all, a certain small group of expressions of this
discipline that seem to us to be immediately understandable; the
expressions of this group we call "PRIMITIVE TERMS" or "UNDEFINED
TERMS," and we employ them without explaining their meaning. At
the samtime, we adopt the principle: not to employ any of the
other expressions of the discipline under consideration, unless
its meaning has first been determined with the help of primitive
terms and of such expressions of the discipline whose meanings
have been explained previously. The sentence which determines the
meaning of a term in this way is called a "DEFINITION", and the
expressions themselves whose meanings have thereby been determined
are accordingly known as "DEFINED TERMS."

We proceed similarly with respect to the asserted statements of
the discipline under consideration. Some of these statements which
to us have the appearance of evidence are chosen as the so-called
"PRIMITIVE STATEMENTS" or "AXIOMS" (also often refered to as
"postulates", but we shall not use the latter term in this
technical meaning here); we accept them as true without in any way
establishing their validity. On the other hand, we agree to accept
any other statement as true only if we have succeeded in
establishing its validity, and to use, while doing so, nothing but
axioms, definitions, and such statements of the discipline the
validity of which has been established previously. As is well
known, statements established in this way are called "proved
statements" or "theorems", and the process of establishing them is
called a "proof". More generally, if whithin logidc or mathematics
we establish one statement on the basis of others, we refer to the
process as a "derivation" or "deduction", and the statement
established in this way is said to be "derived" or "deduced" from
the other statement or to be their "consequence".

Contemporary mathematical logic is one of those disciplines which
are constructed in accordance with the principles just stated[.]

[...]

The method of constructing a discipline in strict accordance with
the principles laid down above is known as the "deductive method";
and the disciplines constructed in this manner are called
"deductive theories" [footnote omitted]. The view has become more
and more common that the deductive method is the only essential
feature by means of which the mathematical disciplines can be
distinguished from all other sciences; not only is every
mathematical discipline a deductive theory, but also, conversely,
every deductive theory is a mathematical discipline (according to
this view deductive logic is also to be counted among the
mathematical disciplines). We will not enter here into a
discussion of the reasons in favor of this view, but merely remark
that it is possible to put forward ponderable arguments in its
support.


Note in particular:

"we distinguish [...] a certain small group of expressions of this
discipline[;] the expressions of this group we call "PRIMITIVE
TERMS" or "UNDEFINED TERMS," and ->we employ them without
explaining their meaning<-."

And

"Some of these statements [...] are chosen as the so-called
"PRIMITIVE STATEMENTS" or "AXIOMS" [...]; ->we accept them as true
without in any way establishing their validity.<-"

Primitive terms are "employed without explaining their meaning", and
axioms are "accepted as true without in any way establishing their
validity".

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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