Proof That Zero Is the First Natural Number (Peano Postulate 3)

An explanation of the proof of the third Peano Postulate, that Zero is the first natural number.
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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!

[@tomholroyd7519]

Can you start with the Peano Axioms and derive Set Theory? We can make sets of natural numbers

[@CarneadesOfCyrene]

The Peano Postulates can be proven from set theory (that is what we are doing here). I don't know of anyone that has been able to do it the other way around.

[@BenjaminBrady-q2g]

If by set theory we mean ZFC, then no. ZFC can form a model (ℤ, +) of Peano arithmetic and prove it to be consistent. Therefore, ZFC is strictly stronger than Peano arithmetic and cannot be encoded. If we think about it, it would intuitively be impossible to even prove something as elementary as the axiom of extensionality (on e.g. equivalence classes of classes over the numbers in Peano arithmetic) without adding an equivalent axiom. Hope this helps

[@hydrogeniongradient3295]

Sir, I thought you have already made a video about that. 🤔

[@CarneadesOfCyrene]

I made a video on the Peano postulate that says zero is a natural number. This is the one showing it is the first.