1. 2 = # {0, {0}}

Sure this can be stated as defining a number? Yes if you are able to take it as just symbols on the RS. Hewever we need to understand the symbols, which requires understanding # as "the number of the set" and to connect 2 with the RS we need to consider # as an operator instructing us to count the sets in sets. But counting relies on numbers already existing.

0 is a number
1 = 0'
2 = 1'
ect.

suffers from the same problem (counting in 1's before 1 is defined).

With these definitions it seems we may need to admit circular definitions for the primitive entities to operate on.

2.

3. This is called a recursive definition. If you define your terms carefully, a recursive definition need not contain circular elements.

4. Best definition of recursion I've ever read in a dictionary:

Recursion = see Recursion

:wink:

5. Originally Posted by talanum1
2 = # {0, {0}}

Sure this can be stated as defining a number? Yes if you are able to take it as just symbols on the RS. Hewever we need to understand the symbols, which requires understanding # as "the number of the set" and to connect 2 with the RS we need to consider # as an operator instructing us to count the sets in sets. But counting relies on numbers already existing.

0 is a number
1 = 0'
2 = 1'
ect.

suffers from the same problem (counting in 1's before 1 is defined).

With these definitions it seems we may need to admit circular definitions for the primitive entities to operate on.
The definitions are not circular, but they do rely in the axiom that each number has a successor. What you are apparently dealing with is the construction of the various number systems starting with the Peano Postulates. See Foundations of Analysis by Landau for a very terse, but efficient, treatment.

Doesn't the understanding of "a number has a successor" requires an understanding of the concept: "indevisable quantum"?

Then

a) 0 is a number, b) each number has a sucessor and

1 = 0' is understood as:

the cypher 1 comes from the number 0 add an indevisable quantum, then 1 becomes a number (by b.). But "an" is replaceable by "one" or "one member of a group" or "one non-specific member of a group" and I doubt if "an" can exist wihtout this possible replacements, and it doesn't have others - check your dictionary.

Please quote the relevant definition if there is a better one.

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