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I was reading elementary set theory and in this book the axiom of union is stated, guaranteeing the existence of some set, say $S$, such that every element of it is in a set $A$ or in a set $B$ where $A$ and $B$ are any sets. Then, the author gives a definition for "union between sets". Such a definition raised the question on the tiltle. From an axiomatic point of view: Is it mandatory substantiate the existence of some mathematical object firstly in order to define it afterwards?

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No. The author could as well have defined the union between two sets first, then to state the axiom of union, and after that to observe that, thanks to that axiom, the union of any two sets always exists.

José Carlos Santos
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  • Cheers! So by knowing is not mandatory, is there a reason to guaratee the existence of some mathematical object firstly in order to define it afterwards, then? Not really? –  Aug 24 '21 at 01:31
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    I don't see why that should always be done in that order.. – José Carlos Santos Aug 24 '21 at 01:32
  • May I ask you a question? Can we define subtraction in $\mathbb{N}$ as follows: $a+b=c\iff b=c-a$ where $a$ and $c$ are arbitrarily chosen natural numbers ($0\in \mathbb{N}$)? One can prove $a<c\implies \exists b\in \mathbb{N}:a+b=c$ such a number $b$ is unique so, does this definition work even though one can also prove $b=c-a\notin \mathbb{N}$ when $c<a$ –  Aug 24 '21 at 01:42
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    If we define $c-a$ as a number $b\in\Bbb N$ such that $a+b=c$, then the definition makes sense, but then $c-a$ is undefined for certain pairs $(a,c)$ of natural numbers, such as $(1,2)$. – José Carlos Santos Aug 24 '21 at 01:47
  • So, does a definition not make sense when does not "cover",in this case, all possible pairs of natural numbers? Why wouldn't it make sense? Trying my hardest to grasp it :( –  Aug 24 '21 at 01:48
  • When does a definition make sense (in an intuitive way)? Please help me :( –  Aug 24 '21 at 01:50
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    At no point I wrote that the definition doesn't make sense in some cases. But what is being defined here is an operation and I wrote that this operation is undefined for certain pairs of numbers. – José Carlos Santos Aug 24 '21 at 01:51
  • Let us continue this discussion in chat. –  Aug 24 '21 at 01:53
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    dear @HannyBoy, whether or not a definition is intuitive, or makes sense, is entirely subjective! it will vary from person to person. if you want to define subtraction in $\mathbb{N}$, you just specify that it is an operation defined only on a proper subset of $\mathbb{N}^2$, and not all of $\mathbb{N}^2$, and then things will be fine – Atticus Stonestrom Aug 24 '21 at 02:10
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Consider the following definition: an odd perfect number is an odd positive integer that is equal to the sum of its proper positive divisors.

A lot of properties have been proved about odd perfect numbers.

But there is one thing we haven't been able to prove: the existence or non-existence of odd perfect numbers. (It's actually conjectured that there isn't odd perfect numbers).

So no, you don't need to prove the existence of an object to define it, study it, prove dozens of theorems about it, etc.

jjagmath
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