What is Peano Arithmetic?

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An explication of Peano Arithmetic, including the five Peano Postulates, the Axioms of Addition and Multiplication, the use of recursion to define the natural numbers, an explanation of mathematical induction as well as several proofs using these axioms and rules.

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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 and more!

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 and more!

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It's amazing how much time it really takes to actually grasp all the notions and see the complete idea in two different ways (plain Peano arithmetic with or without using set theory).

Knowledge plays a great drama. The thing that we're trying to arrive to, is always in front of us, but we don't see it. With time, when we see it, we see ourselves already arrived to it. Now, the world stays the same, but makes more sense.

This is a struggle for over a year. My mind is shouting "Eureka" but can't say what to anyone else unless they already had this experience but those who know already knows.
I've something so significant in my hands but can't share it anyone other than those who have it. woah the cycle repeats... :)

Btw, thanks for the nice video. Appreciate your time & effort in making these :)

deepaks.m.

Thank you. This what a great video for me to learn how Peano constructed arithmetic.

I have some issues with his reasoning and method, and feel there are ways to construct arithmetic from sound premises, no bold assertions (axioms) that are simply taken. Sound axioms, so to speak. Axioms derived from inherent truths that we are aware of from primal notions (q.v.), such as concatenation, ontological necessity, and... the existence of unity. Not zero as Peano did.

After all, we didn't start with zero and lived a while without it. It is more basic if when asked (after a counting system was developed), "How many goats do you have?", for someone to respond with, "I do not have any goats", or simply "not any". As in, "no such thing exists", an absence from existence. So we have unity and possibly a concept of a null set.

Successorship is great, but needs to be defined better. It isn't ligometric to state the axiom as he did.

Anyway, I learned a lot about how Peano thought, and this certainly helped. Thanks again.

snatchngrab

This video is great! But how would we define the successor function? How do we know which member of the set of natural numbers it will map onto? Who's to stop someone from saying S(o)=3, S(2)=6, and simply ignore every number that's not a multiple of three?

We seem to just take successorship for granted without explicitly defining S(o)=1 and SS(o)=S(1)=2.

Also, a related question, how is the order of the set of natural numbers defined (other than zero being the first member of the set)?

Is there something I'm missing in the rigour? Or should some of these notions be axioms or postulates in their own right?

TheGeneralThings

Well done. Good video! Did you use principia mathematica? (BERTRAND RUSSELL)

maskedmarauder

Are there any other additions to second-order logic than a universal and existential quantifier over predicates and isomorphs of deductive rules regarding these qualifiers from first order predicate calculus?

jakehaun

Hi. how can i progress in mathematics ? Do you have methods or something ? Now, I understand the basics pEano properties. But How can i think to demonstrate it? Thank you.

waldoarmagon