What is Zermelo Successorship
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- čas přidán 2. 03. 2024
- An explanation of the Zermelo framework for successorship where the successor of a set is just the set of that set.
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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, The Oxford Companion to Philosophy, The Routledge Encyclopedia of Philosophy, The Collier-MacMillan Encyclopedia of Philosophy, the Dictionary of Continental Philosophy, and more! (#SetTheory #Zermelo)
Could you please talk about the axiom of equality 🙏
isn't this basically just another variant of Church encoding?
You should do one on minority rights. I believe utilitarian and deontological ethics alike protect the rights of minorities. “The greatest good for the greatest amount” isn’t “the greatest good for the greatest amount, unless you are disabled. Unless I’m wrong, and certain utilitarian arguments actually support the eugenics movement. We should also look to natural law and natural rights arguments.
If we can "count" the number of brackets, we can count other things - like apples. If we can count apples, then we simply have number of apples. No sets necessary.
yeah but the point here is defining formally what a number is, because the aim of set theory is to serve as a foundation for all mathematics.
you will see that defining numbers as sets will have its advantages (especially when talking about infinities, set theory is the ideal framework)
@@ostrich_dog what I am saying is that you can't "count" without numbers already defined. If we can count the number of brackets, we could just as easily count the number of apples.
@@InventiveHarvest surely the way we formally build natural numbers is influenced by our pre-existing intuitive notion of natural number.
however the "counting" of the brackets isn't really used to distinguish a number from each other, that's just a way for us to interpret the syntax, whereas the sets themselves are distinguished just because they have a different element, and their order comes from the successor relationship that holds between any two consecutive numbers, while the number of brackets is just a consequence of that.
we might say that the set theoretic definition, although it comes "logically" before our intuitive notion of counting, comes "psychologically" after that, but that isn't a problem because what we really care about in this context is just the logical way in which set theory establishes a foundation
@@ostrich_dog how is counting logical when you don't have numbers?
@@InventiveHarvest counting is itself logically defined by the application of the successor operation to a certain number. we first define 0, and then define all the other numbers through this form of counting.
but there is a little ambiguity, because when he says "count the brackets" he doesn't mean "count" according to the logical definition, but according to our intuitive understanding of counting that we already had even before these formalities, which are two separate things: you are correct in saying that we need to already know what a number is before talking about sets, but we can also find a way to talk about sets which, although we understand thanks to our intuitive notion of number, is independent of it
One question: does this have an value or application? It seems to me that in order to use set of a set of a set to count to 3 you first have to be able to count sets? Isn't this circular? if 5 is {{{{{null}}}}} then you need to count the sets?