Jim Burns wrote: > On 8/16/2013 4:54 AM, Peter Percival wrote: >> Can addition be defined in terms of multiplication? I.e., >> is there a formula in the language of arithmetic >> >> x + y = z <-> ... >> >> such that in '...' any of the symbols of arithmetic >> except + may occur? Or, alternatively, is there a >> formula in the language of arithmetic >> >> x + y = ... >> >> with the same requirement? >> >> The symbols of arithmetic (for the purpose of this question) are either >> >> individual variables, (classical) logical constants including =, >> S, +, *, and punctuation marks; >> >> or the above with < as an additional binary predicate symbol. > > How about > > x + y = z <-> 2^x * 2^y = 2^z > > where 2^x is just an abbreviation for the function 2pwr: N -> N, > defined by > 2pwr(0) = 1 > > 2pwr( Sx ) = 2 * 2pwr( x )
That goes beyond what I defined as the language of arithmetic.
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