What is the Principle of Mathematical Induction (Peano Postulate 5)

An explanation of the fifth and final Peano Postulate, the Principle of Mathematical Induction, which claims that if zero is in a set and if being in a set implies that your successor is in that set, it means that all natural numbers are in 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 #Peano)

[@roderictaylor]

Very nice!
The principle of mathematical induction also excludes the model for the natural numbers:
0, 1, 2, 3, 4, 5, . . .
.. . . .infinity-3, infinity-2, infinity-1,infinity, infinity+1, infinity+2, infinity+3, . . .
Mathematical induction encapsulates our intuition that every natural number is finite, in the sense that every natural number can be reached from 0 by applying the successor function a finite number of times.

[@kenoctcercos4832]

Could you please do also a video on plural logic and plural quantification?

[@Dayglodaydreams]

I’m going to follow Wes Cecil and Dr. Gregory Sadler for six months.

[@InventiveHarvest]

Seems problematic. We are using these postulates to define "natural numbers". But the postulate has the clause "for all natural numbers" in it, which apparently hasn't been defined yet.

[@CarneadesOfCyrene]

A good point. "Define" is probably the wrong word here. These postulates do not so much define natural numbers as they model them. In other words, they show that if a group of things satisfies these postulates, then those things are a perfect model for the natural numbers. We are going to offer definitions of the three primitive concepts (Zero, Successorship, Natural Numbers) in terms of set theory, then prove that these postulates all apply to the sets that match those definitions.