Proof that Zero is a Natural Number

A step by step proof of the first Peano Postulate, that Zero is a 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! (#settheory #peano)

[InventiveHarvest]

What they told me in grade school was pretty similar. The definition of the natural numbers is all of the whole numbers and zero. Therefore, zero is a natural numbers by definition.

[icewlf]

hopefully no one uses this video to disprove someone who takes the 1 as the smallest natural number, since the answer here depends on how you defined inductive set (weather it "must include 0" or "must include 1")

[wandrespupilo]

wait, that argument seems to me to be a bit circular...
before ZFC, the DEFINITION of the natural numbers (as it was tought to me in grad school) were the peano axioms, so when building the numbers with sets, what is done is to build them SUCH THAT you can prove the postulates, therefore proving that those sets are actually natural numbers (what you said implies that the REASON for the postulates are the numbers as sets, and i disagree). This could be the case if ZFC came before Peano arithmetic

[patrickwithee]

Around the 4 minute mark, why don’t you just assume IN(B) for an arbitrary B, and then just use the definition to derive 0€B, followed by universal intro to get the desired result? Just curious.

[CMVMic]

So we prove zero is a natural number by defining it as a natural number?

[manuel]

Why would you define zero as equal to the null set???