This discussion addresses the proof of the non-existence of a bijection between a natural number set $$x$$ and its successor set $$x^+$$, defined as $$x^+=x\cup\{x\}$$. The conclusion is that the cardinality of $$x$$ is less than that of $$x^+$$, formally stated as $$\mathrm{card}\,x<\mathrm{card}\,x^+$$. The argument relies on the properties of natural numbers and the Axiom of Infinity, as discussed in Zorich's "Mathematical Analysis". The assertion that $$x$$ is infinite leads to a contradiction, reinforcing the proof's validity.
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I took this problem from Zorich "Mathematical analysis". Zorich spoke just twice about infinity before this problem: 1. Dedekind's definition, 2. Axiom of infinity. Suppose there is a bijection from $$x$$ to $$x^+.$$ Since $$x\subset x^+$$, then $$x$$ is infinite by 1, which is false, but I don't know how to prove it.Andrei said:$$\mathrm{card}\,x<\mathrm{card}\,x^+.$$