First order of logic - x is a power of 2

Asked Oct 30 '20 at 17:09

Active Nov 03 '20 at 13:56

Viewed 170 times

I don't think something like (we can assume that the formula is over $\mathbb{N}$):

$\exists y. 2^y=x$

is allowed (it may be - I am just not sure).

I don't see a way to do it recursively without reaching an infinite formula.

Can anyone help please?

Edit: Using power in the formula is not allowed, and according to my professor this should be done using division and the unique prime factorization of a number.

edited Nov 03 '20 at 13:56

asked Oct 30 '20 at 17:09

EL_9

does that mean 'for every $y$ there is an $x$ such that $x = 2^y$' ? – Spectre Oct 30 '20 at 17:11
@Spectre No? It means that there exists $y$ so that $2^y=x$ – EL_9 Oct 30 '20 at 17:12
1

Does this answer your question? How is exponentiation defined in Peano arithmetic? – Natalie Clarius Oct 30 '20 at 17:57
@lemontree Thank you. The solution proposed in the second answer seems too complicated for the level we were taught at. Is there a simpler way? – EL_9 Oct 30 '20 at 18:50
If there were a known simpler way, you can be assured Hagen and Andrés (whose answer expands on and explains Hagen's) would have given it. – Paul Sinclair Oct 31 '20 at 00:48
@PaulSinclair I asked the professor and he said that we should use division, and the unique prime factorization of a number. – EL_9 Nov 03 '20 at 13:55
Ummm What exactly is the questioh you are trying to answer?? You have never made that clear, and it now appears that no one who has responded so far has been correct in our guesses. – Paul Sinclair Nov 03 '20 at 17:09

First order of logic - x is a power of 2

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