Set Theory

Whenever I peeked into the confounding universe of set theory, the infinite and the continuum, I would be utterly lost in looping thoughts and would only be rescued by sleep (which would inevitably be accompanied by strange dreams), or callings of daily schedule. Thus saved, I would brush off my hands and stay clear from any such potentially paradoxical digressions for as long as possible before plunging into it yet again and pushing my luck further..

lo and behold! this time I actually covered some new ground. Very simply: if we analyse the statement ‘All A’s are B’s”- By this do we mean that all A’s known to us and that can be seen, distinguished or counted and included in the class A – all such A’s are in B? Or do we mean that any entity having the essence or property of belonging in class A will also be in B? For example.. “All triangles contain two right angles” is it true because it has been shown true for all possible triangles (by summing up the angles) or its it true because it is an essential property of being a triangle? The former is known as the extensional reading while the latter, as theorized by Aristotle in terms of ‘essence’ and ‘kinds’ of things is intentional reading. As it usually happens, applied mathematicians fudged over these details and went ahead with their algebra, while it was left to the camp of philosophers (and some queer mathematicians) to come up with consensus.

Alas, its not only me who is baffled by infinities. The first time I came across Zeno’s paradoxes I could not stop thinking for many hours about how we actually manage to move (there are infinite no. of points in between any two steps). And I had some sort of respect for life you know.. transcending infinities in space.. and so on.. my poetic renderings of the phenomenon. But the initial hurdle has been obviated, thanks to philosophers again, what with whole being different from its composite parts, potential infinity versus actual infinity and so on…

And now I discovered another distinction.. between being finite and having boundaries. For some reason we tend to think that everything that is finite in scope must have a boundary. But what about circumference of a circle or surface of Earth? Where is the starting and ending point?

More on this later (hopefully)…

(Reading Philosophy of Set Theory by Mary Tiles)

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