What troubles me is the notion of equality and related axioms. I used to think that “x=y” means that “x and y are precisely the same thing”. But now it seems to me that the statement “x and y are the same object” is actually defined by the statement “x=y”,which seems more formal in mathematics. And the relation of equality, to a large extent, is just a matter of definition. We may define the relation of equality for a class of objects freely so long as the axioms of equality(reflexivity, symmetry, transitivity and the axiom of substitution) are all satisfied. Is that correct?
What troubles me most is the axiom of substitution, though. I know that any operation on objects defined using previous well-defined operations on those objects is also well-defined. Sometimes I succeed in understanding this (For instance, subtraction on the real numbers is well-defined, because subtraction is defined in terms of previous well-defined operations, namely addition and negation), but other times, I fail to understand this. For instance, suppose that m and m’ are integers such that m=m’, and suppose that $(x_n)_{n=m}^\infty$ is a sequence of reals with starting index m,while $(x_n)_{n=m’}^\infty$ is a sequence of reals with starting index m’.Then here are some questions: 1. if $(x_n)_{n=m}^\infty$ converges to some $L∈\mathbb{R}$, does $(x_n)_{n=m’}^\infty$ converge to $L$? 2. does the equation $sup(x_n)_{n=m}^\infty=sup(x_n)_{n=m'}^\infty$ hold? 3. if $(x_n)_{n=m}^\infty$ is a Cauchy sequence, is $(x_n)_{n=m'}^\infty$ also a Cauchy sequence? ... ...
I know that these propositions are super easy to prove, but my question is, do we actually need to prove them ?(Think about the axiom of substitution for subtraction mentioned above.Though we can prove it step by step, we don’t need to)
Is it meaningful to think of these kinds of questions? (It seems I have been trapped by these questions.) I find that in most textbooks on analysis or algebra, the authors never talk about the axiom of substitution.
I'm not a native speaker of English, and I apologize for any possible misunderstanding caused by the language barrier.
P.S. Let me add a proof here. Suppose that $(x_n)_{n=m}^\infty$ is a Cauchy sequence, then let's prove that $(x_n)_{n=m'}^\infty$ is also a Cauchy sequence. Since $(x_n)_{n=m}^\infty$ is a Cauchy sequence, we have $\forall ε＞0:\exists N≥m:\forall j,k≥N:|x_j -x_k |＜ε$. But we know that $m=m'$, then certainly we have $\forall ε＞0:\exists N≥m':\forall j,k≥N:|x_j -x_k |＜ε$, which means that $(x_n)_{n=m'}^\infty$ is Cauchy. The key to this question is the substitution property for $≤$，i.e., $(N≥m \land m=m')\to(N≥m')$.
