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the symbol 0 has many uses. 0 could be the identity element in a group.

0 could be a cardinal number or it could be a natural number. in the first case, |{}|=0 and in the second case, {}=0. it can be ambiguous that 0 is used so much. as far as i know, |{}|!={} yet people use the symbol 0 for both.

while this construction of N isn't necessarily "wrong," i like the ZFC better because it doesn't mention cardinality (under the general impression that less is more).

0 could be a cardinal number or it could be a natural number. in the first case, |{}|=0 and in the second case, {}=0. it can be ambiguous that 0 is used so much. as far as i know, |{}|!={} yet people use the symbol 0 for both.

of 1={0}, then shouldn't it be 1=||{}|| and not |{{}}|?0 = |{ }| (notation = {})

1 = |{{ }}| (notation = {0})

while this construction of N isn't necessarily "wrong," i like the ZFC better because it doesn't mention cardinality (under the general impression that less is more).

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