PEANO Integers using successor function

Question

I'd like to define the integers and operations on it by using terms. Starting from zero->init[] and nth->succ[n-1th] where succ[] is the successor function. I believe the operations could be implemented as :

add[succ[x],y]=succ[add[x,y]]
product[succ[x],y]=add[product[x,y],y]

But how can I actually define the set or build a successor function?

Thanks

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score 3 · Accepted Answer · answered Dec 19 '14 at 04:05

It depends on how raw you want to go. Here is an approach that maintains the symbolic character of the formalism:

ClearAll[convertToS, convertFromS, add];

convertToS[0] := 0;
convertToS[n_] := Last[ComposeList[ConstantArray[S, n], 0]];
convertFromS[chain_] := Count[chain, S, Infinity, Heads -> True];

add[a_, 0] := a;
add[0, a_] := a;
add[chain1_, S[chain2_]] := add[S[chain1], chain2];

five = convertToS[5]
sixtyFour = convertToS[64]

add[five, sixtyFour] // convertFromS

S[S[S[S[S[0]]]]]

S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[S[0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

69

Here the convertToS and convertFromS functions are "outside" the system. Addition just involves commuting all S symbols to the chain on the left. Offhandedly I'm not quite sure how the product could be defined, but it could be done through copying. Philosophically though, in trying to define the numbers you can't escape the usage of them in the first place, implicitly or otherwise.

bill s · Answer 2 · 2014-12-19T03:31:58.713

According to this Wiki article, the successor function is the function with S(1) = 2 and S(2) = 3.. this can be implemented in several ways, perhaps the simplest of which is

s[n_] := n + 1

Hence s[2] returns 3 and s[10] returns 11.

The add function is handled automatically since s[n]+m is 1+n+m. The product is similar since, for instance, s[n] m is automatically simplified to m (1+n). But I'm not sure I understand what you are really trying to do.

PEANO Integers using successor function

2 Answers2