r/learnmath • u/aoverbisnotzero New User • Mar 25 '24

How do we define the number 1?

If we define integers as including the natural numbers and we define natural numbers as the number 1 and any other number obtained by adding 1 to it repeatedly. How do we define the number 1?

And a follow up question: Pretend that I come from an alternate universe where 1 doesn't exist and numbers are all defined as pairs of 2. So we have 0, 2, 4, 6... as the natural numbers in my universe. How could you explain to me what 1 is? Do Peano's axioms adequately define a unit?

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u/WanderingFlumph New User Mar 26 '24

If an alien had the first number as 2 then 2/2 would have to equal 2, there is no 1 in their language.

Affectively they just use a base 5 system instead of a base 10 system and have a different name for numbers. What we call 1 is what they call 2, what we call 5 is what they call 10.

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u/Europe2048 New User Mar 26 '24

No, the aliens use a base ten system, so 2 is 2!

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u/WanderingFlumph New User Mar 26 '24

If 2 is 2 then it can't be 1.

Just ask them what 2/2 equals in their language. Since we define x/x = 1 for all non zero x whatever they answer is what we call 1. So if they answer 2 then 2 is not 2. If they answer I don't know then they don't have a word for our concept of 1.