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MATE5416
Set Theory
5–10
ECTS

Periods

Language of instruction

English

Type or level of studies

Advanced studies

Course unit descriptions in the curriculum

- MATE5416 Set Theory

Matematiikka

Department of Mathematics and Statistics

Department of Mathematics and Statistics

**Set theory and Mathematical Practice-**

**Finite computability, measure, the continuum**

We will give introduction to standard set theory but by stressing two nonstandard themes (that in Cantor's hands--the founding father--were one)

1. we emphasize the historical emergence of set theory from complex analysis in between 1870-1880

2. we stress throughout the course the critical--often necessary uses--of "abstract" set theory in settling simple propositions about the real line concerning measure theory, features of real functions, how many points are there in this or that set of reals etc.

It is commonly assumed much of ZFC(standard axiomatic set theory) is not "needed" in ordinary math practice. We show that this is not quite right-- (very) large cardinal theory is essential. The theme goes back to Friedman/Martin result of the 1970's about Borel determinacy but today there are many more such examples. We connect this to use of infinity axioms in gauging "consistency results".

We will study Cantor's 1883 informal principles; then his 1899 sets vs. classes; the well ordering theorem (axiom?).Then Zermelo's 1908 axiomatization ZC, without the axiom F and in a full second order language; then Skolem's modern formal version (1920-2) with the axiom F in first order form + denumerable models for set theory (and the so called Skolem Paradox). We will reach Godel's key minimal model L (via Von Neumann's earlier mini models). The last segment will look at recent work of Hugh Woodin bent on settling Cantor's 1878 continuum hypothesis in a revised form of Godel's L, called Ultimate L, wherein all (yes, all!) large infinities(cardinals) are accommodated.

The course is interested in understanding

(1) why set theory has become a sort of universal language of math

(2) asking: is it a universal language of math?

(3) why are many structural features of R missed by standard set theory?

(4) is set theory fully articulating the notion of ...infinity?

Given the informality of our approach, any student (without fancy background) can start and try. Lectures start September 15that 12 in Quantum (room M3, third floor, Math and Stat Dept.)

Welcome!

Joseph Almog and Vesa Halava

Vesa Halava

vehalava[ät]utu.fi

Joseph Almog

Lectures

Thu 15-Sep-2016 - 8-Dec-2016 weekly at 12-14, M3, Quantum

Thu 15-Dec-2016 at 10-12, M3, Quantum