exfret wrote:alessandro95 wrote:That axiom isn't there specifically to exclude ∞, it is there to state a very useful properties of real numbers without which they simply won't work as we wish them to and which happens to exclude infinitesimals and infinities from the field of reals

Okay, but this only means that infinity is only excluded because it's an oddball and not because and not a number, right?

We decide the axioms first and then we call real numbers the field obtained from the axiom, you're doing things in the opposite order, we're not excluding ∞ because it isn't a number, but we're excluding it because it breaks one of the axiom therefore it isn't a real number.

5+3i breaks the oder relation on R so it's not a real number but it can be a number in another system, as well as "apple" can, if you define operations involving it.

exfret wrote:alessandro95 wrote:R is defined as a field {R,+,*,<}.

Let's not worry about the order relation but only with addition and multiplication for a moment.

The definition of a field is a commutative ring (in this case {R,+,*}), which has a multiplicative inverse for every element but the additive identity, and R contains indeed a multiplicative inverse for number except 0, and it is not supposed to contain one for 0, which is why 1/0 is undefined

I can see that a field is defined for adding, multiplying, and order, (I overlooked that last one) but why do we have to specifically exclude the multiplicative inverse of the additive identity? Even if we should exclude it just so it doesn't violate the real number's properties, why does it have to be excluded from all fields? Also, why must it be undefined just because of this exclusion? It has a value that can be defined. Here: I define x such that x = 1/0. There. Call it meaningless, but it's not undefined. Edit: Call it meaningless but it's not undefined, right?

1/n is the multiplicative inverse of n and since 0 is the additive identity it is defined not to have an inverse or, looking at things the other way around its multiplicative inverse is undefined.

(this is because there isn't a value of x for which 0x=1)

Of course you can say "let x=1/0" but then we are no longer working in a field and we lose a lot of nice properties of fields, there are interesting structures such as wheels where division by 0 is defined but defining the reals as a structure different than a field would bring much more disadvantages than advantages.

exfret wrote:alessandro95 wrote:1/0 is undefined in the fields of reals for the fields axioms I quoted above

So it's undefined for the reals? But that doesn't make any sense. Edit: But that doesn't make any sense to me. "The Reals" are just an isolated part of mathematics that we created mostly because they can be ordered. 1/0 would be inexistent for the reals. Of course, I still don't understand why you'd have such a restriction to use explicitly only real numbers anyways.

You don't have to restrict yourself to real numbers, but when you work with reals you have to follow the axioms of reals and their implications (1/0 is undefined, ∞ isn't a real number and so on), it is perfectly fine to work in another system (you may want to read about the hyperreals numbers if you didn't already do that) but you can't give a value to things such as 1/0 inside the reals.

exfret wrote:alessandro95 wrote:I did tell you, it is because of fields axioms

Oh, sorry. I overlooked that because I was expecting different reasons (e.g. zero times any number is always zero). Anyways, the field of axioms would state that it's undefined for a field because it would be the multiplicative inverse of the additive identity (which is what I said myself), is that correct? But, why would it be 'undefined'? It's a number, a defined number, but just one outside of the specific field one might be focusing on. I find it as absurd as calling i undefined when working with reals. Edit: That's just an analogy, I'm not calling your idea absurd.

Of course it could have a defined value inside another algebraic structure, but you need precise axioms describing it and the operations it posses.

I may have expressed myself poorly yesterday (english isn't my language) but what I meant is that 1/0 and ∞ have no meaning inside the field of reals, because there is no way to assign them a value staying inside the field, but that doesn't mean infinity is a meaningless concept, it means ∞ it's not a number but a symbol (when working with the reals of course) and that we cannot evaluate operations involving it in a meaningful way inside the reals.