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I'm reading David Steward's "Foundations of Mathematics" and in chapter 8 he is building an axiomatic system for the natural numbers with addition defined using Peano's Axioms. I don't really fully understand the need for this. Why can't I just define addition in the same way I learnt it in primary school - using an addition table - but define it for all natural numbers?

Thanks,

drd26

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    $\begingroup$ You can't write down that table, so you can't use it for the definition. In fact you construct the table, or a finite part of it, using some other definition $\endgroup$
    – David C. Ullrich
    Jun 12, 2022 at 10:58
  • $\begingroup$ Thanks very much. Why does not being able to write down a table imply that it can't be used as a definition. And what do you mean by "you construct the table...using some other definition"? $\endgroup$
    – drd26
    Jun 12, 2022 at 11:19
  • $\begingroup$ You could treat all those infinite entries of the table as axioms ... but now you have a problem: if I do a proof using one of those entries as a premise, then it only counts as a proof if you can verify that that premise is in fact an axiom: So I would need to go through the list of axioms and find that premise there. But with infinitely many axioms, this is not something I can do: there is no effective algorithm that can deal with this. So this is why we don't want an infinite number of axioms. We can have axiom schemas that have infinitely many instances, but that's not the case here. $\endgroup$
    – Bram28
    Jun 13, 2022 at 17:11

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Elaborating on a comment. It's impossible to use that table as the definition. Because since the table is infinite you can't write it all down, so you have nowhere to look up the result of 3+17.

Really. definitions have to be finite. You can use a table to define the addition in $\Bbb Z_2$; that amounts to simply defining $$0+0=1+1=0, 0+1=1+0=1.$$Now if you have to add to elements of $\Bbb Z_2$ you can simply look up the result in that display.

But that doesn't work for the infinite table you'd need to add integers. No matter how large a finite part of the table you're given, there is some addition problem that it doesn't cover.

Hmm, look at it this way: If we haven't defined addition it's impossible for you to tell me exactly what "table" you're talking about! You can't write it all down for me, the best you can do is say "the table with $x+y$ in row $x$ and column $y$", and if you say that I won't have any idea what you're talking about, because addition is undefined.

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  • $\begingroup$ The last paragraph is really sweet ;-) $\endgroup$
    – Gottfried Helms
    Jun 12, 2022 at 11:45
  • $\begingroup$ Thanks very much David - that's really helpful. $\endgroup$
    – drd26
    Jun 12, 2022 at 18:31
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    $\begingroup$ @GottfriedHelms Thx. I admit I was kinda pleased with it, a "dramatic" exposition of why it's really impossible, whether we feel like allowing it or not... $\endgroup$
    – David C. Ullrich
    Jun 12, 2022 at 18:54

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