First, I don't think this is set theory proper -- the symbols 0, 1, 2, ... are just abbreviations, with only SSSSSSSSSSSSSSSSSSSSSSS{} (etc.) appearing in the theory itself. But given Peano definitions of addition and multiplication it's not hard to define concatenation as the right-associative operation ab := (10*a) + b. Charles Greathouse Analyst/Programmer Case Western Reserve University On Tue, Nov 11, 2014 at 8:44 PM, David Wilson <davidwwilson@comcast.net> wrote:

In set theoretical (ZF or ZFC) developments of the natural numbers, I have often seen the successor function defined

Sn = n ∪ {n}

and a few of the smaller numerals defined

[1] 0 = ∅, 1 = S0, 2 = S1, 3 = S2, &c.

The &c has no meaning within the theory, the student is expected to expand the &c to an infinitude of like definitions for the remaining numerals. Later, if the student encounters the string 23 within some statement of the theory, he is expected to interpret 23 as shorthand for SSSSSSSSSSSSSSSSSSSSSSS0, even though he was never given neither an explicit definition of the numeral 23 nor a general definition of numerals whereby he could deduce the definition of 23.

My question is, can the set theoretic development of natural numbers be augmented with a set theoretic definition of numerals so that if 0 or 23 or 1093 or 6661661161 or any other string of digits has a definition as a natural number within the theory, without recourse to the student’s understanding of the values of the numerals?

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