Per Helders: Richard's paradox: Impredicativity and infinity coinciding
Tid: Fr 2020-12-11 kl 10.00 - 11.00
Föreläsare: Per Helders
The principal purpose of this paper is to, on the basis of Richard's paradox, study how mathematicians over time have discussed some paradoxical reasonings which are more or less connected to that paradox. Hence, we will look upon these questions from a partly historical perspective, but also treat the question why some of these paradoxical reasonings are still around and discussed.
However, when studying this issue, an additional issue emerges, and with that issue also an additional purpose. This additional purpose is to study, with mathematical/logical tools, the issue of the split or discrepancy between classical and constructive/intuitionistic mathematics. This split is so closely related to some of the paradoxical reasonings mentioned above, that it can be explained in terms of the principal questions raised, and, furthermore, the location and the nature of this split can be illuminated by the principal results. Therefore, we will make a synoptic account, both historically and in terms of content, of the split mentioned. Doing this, we will initially focus on two mathematical topics that have been, and still are, regarded as problematic. It is inside the framework of these two topics that answers have been sought for concerning problems revealed by some paradoxical reasoning. The two topics are:
- Impredicativity (circularity)
- Quantification over infinite sets
We will see that Richard´s paradox, maybe more than any other, reveals problems belon- ging to these two topics, not in the sense of problems as being part of the paradox, but rather as being part of the mathematical reasoning from the time of its presentation and up until today. Indeed, Richard´s paradox is the event where these two topics coincide in a way that is very useful for an analysis of them and of the contradictory reasonings connected to them (i.e. is there a common denominator?), but also, as we will show, calls for caution to carefully distinguish between these two topics when they coincide. Additionally, as a third topic, the analysis will also have impact on what ontological status we assert to different kinds of mathematical objects.