The Transition to Set Theory

It is the current thesis of many mathematicians that set theory has now replaced geometry as the ultimate model for rigorous thought. They may be correct. Through set theory quite a number of problems have been resolved, and with derivatives of set theory such as model theory issues of our language itself can be shown to create many new problems that can be fully analysed within the context of language models.

Abraham Robinson (1918-1955)

Abraham Robinson was born on October 6 1918 in Waldenburg, Germany. Robinson's family emigrated to Palestine in 1933. There Robinson studied mathematics under Fraenkel. During World War II, he worked on aerodynamics. After the war studied at London University, receiving his PhD in 1949. The area of his dissertation was model theory. (Classical model theory, now called first-order model theory, is a branch of mathematics that deals with the relationships between descriptions in first-order languages and the structures that satisfy these descriptions. A first-order language is formed by building up formulas from the symbols in the signature, where the signature is a set of individual constants, predicate symbols and function symbols; each of the predicate symbols and function symbols has an arity (for example it is binary if its arity is 2). For example the field of real numbers forms a structure whose elements are the real numbers, with signature consisting of the individual constant 0 to name the number zero, a 1-ary function symbol - for minus, and two 2-ary function symbols + and $\times$ for plus and times. )

He is most famous for his contributions to what is called nonstandard analysis, a variant of real analysis, which allows actual infinitesimals and infinities --- without paradoxes. For example, we can say the if $|x_{n}-a|$ is an infinitesimal for every infinite value of . Achieving such a mathematics had long been sought but without success. Among the many contributions of nonstandard analysis is a completely rigorous calculus. There have even been written elementary books on calculus based on nonstandard analysis. While a powerful tool, nonstandard analysis has not caught on. In this brief chapter, we will consider some philosophical observations of Robinson.

Robinson died of cancer on April 11, 1974 in New Haven, Connecticut.
MATH

Philosophical comments

Thesis: The results of the centuries arouse deep admiration but are built on shifting sands. of foundations of mathematics now as well as in the past.

Commonly accepted: math as a deductive science begin with Greeks 4 $^{\QTR{rm}{th}}$ --5 $^{\QTR{rm}{th}}$ century B.C.
What about the Babylonian? the Egyptians? Was their mathematics not deductive?
What would the Babylonian mathematician say of Euclid?
Major discoveries were not the \underline{invention} of new math but the \underline{codification} of elements of mathematical thought -- explicit arguments notions assumptions rules etc. which had been intuitive for a long time.

Thesis: The differentiation between \underline {Philosophy} and \underline{Mathematics} was less distinct then than now.

THEN: One could know both.

NOW: It is very difficult to be dually conversant.

Hence -- Plato or Aristotle or some other philosophical school could have influenced the development of the Axiomatic method even though none that we know of (except Democritus (atomism and math treatises) contributed to mathematics.
Aristotle established standards of rigor and completeness which went far beyond the level actually reached at that time.
Euclid and Archimedes singled out those axioms which could not be taken for granted and then proceeded to produce them with assumptions whose truth seemed obvious -- by means of rules of deduction seemingly equally obvious.
Yet Euclid did make the Archimedian Axiom as a definition because he didn't wish to exclude the other. He needed it for the theory of proportion and method of exhaustion.

Thesis: Axioms have always (until early 19 $^{\QTR{rm}{th}}$ century) been regarded as statements of .

However in Euclid there is an element of "constructivism" which should hark back to pre-Hellenic times and should strike a chord in the hearts of those believing Mathematics has been pushed too far in a formal-deductive direction and needs a more constructive approach in the foundations.
Ironic fate: only after Geometry lost its standing as the basis of all Mathematics was its axiomatic foundations finally reached the degree of perfection which the public estimation had always given them.

Soon thereafter the first form axiomatic theories were proposed.

20 $^{\text{th}}$ century:

Set theory achieves the position once occupied by Geometry.

Set Theory also went through a similar evolution.

The initial assumptions of Set Theory were held to be intuitively clear being based on natural laws of thought
Then Set Theory was put on a postulated basis including the axiom of choice (Zorn's lemma) the least intuitive At this point the axioms were still suppose to describe reality, although Platonic.
Then came the realization that it is equally consistent to deny or affirm some major assertions of Set Theory e.q. continuum hypothesis mid 1960s. This led to the end of Set Theory as describing objective reality.

Set Theory and Mathematical Logic

Advances have been made through the codification of notion (e.g. truth). There is every reason to believe the codification will continue, bringing new advances.

References:

Dictionary of Scientific Biography
A J Macintyre, Abraham Robinson 1918-1974, Bull. Amer. Math. Soc. 83

(1977), 646-666.
A D Young, S Kochen, S K-rner and P Roquette, Abraham Robinson, Bull.

London Math. Soc. 8 (1976), 307-323.
J W Dauben, Abraham Robinson : The Creation of Nonstandard Analysis,

a Personal and Mathematical Odyssey (Princeton, 1995).
G D Mostow, Abraham Robinson 1918-1974, Israel Journal of Mathematics

25 (1976), 4-14.

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	THEN:	One could know both.
	NOW:	It is very difficult to be dually conversant.