CrankFan said:
You've not given any justification for your belief that the foundation of mathematics, as it's currently understood, is insufficient or inconsistent. Instead, you've insisted that some facet of the current approach is wrong (like say, defining 0 first) and fail to show how it leads to either a contradiction or limitation in characterizing some useful area of mathematics. Essentially, you don't like construction of N starting from 0, but can't articulate why this is problematic; it's just a (religious) belief.
With all due respect it wasn't my intent to prove or show anything. I was merely responding to the original poster's concern with proving addition. I simply providing information concerning that topic.
I think that there's been a general consensus among the responders to this thread that addition cannot be proven (or as some claim it is even meaningless to ask such a question). I agree that it cannot be proven using current mathematical axioms. And I also agree that within that axiomatic framework it is a nonsensical question.
I was merely pointing out that the particular axiomatic framework that is preventing the question from being meaningful was introduced rather recently (with respect to the entire history of mathematics). And that the theory that prevents it is Georg Cantor's empty set theory.
I believe that everything that I've said here is true. I believe this on solid logical grounds. I actually don't have a religious bone in my body. (maybe spiritual, but that's another topic)
I have no intent to attempt to convince people who aren't interested in this topic. The mathematical community as a whole is well aware that there are logical problems associated with set theory. This is no secret and it has been a philosophical debate for many years (not the specific empty set concept that I am referring to, but set theory as a whole logical system).
Just for a quickie I will give you concrete example of the problem,…
Originally (that is to say that before the time of Georg Cantor) a set was intuitively understood as a collection of things. Cantor then introduced the idea of an empty set. A set containing nothing.
Well, this is actually a logical contradiction right here. We have a collection of things that is not the collection of a thing. This is a logical contradiction to the very definition of what it means to be a set. Similar objections actually came up by other mathematicians at the time that Cantor proposed his empty set idea. These objections were never (and I mean NEVER) fully resolved. In fact, the significance of their implications wasn't even fully understood at the time. Nor does it seem to be fully understood today.
In any case, there are only two ways to get around this logical contradiction. One is to claim that nothing is a thing in its own right. Therefore the empty set does indeed contain a
thing and there is no logical contradiction. However, that logic leads to further contradictions by the simple fact that they empty set is then no longer empty. It contains this thing called
nothing. In fact, this solution was pretty much tossed out as being far too problematic. The so-called "genius" of Cantor's idea was to remove the idea of number from any connection to the idea of a thing thus making it a "pure" concept. Even Cantor did not like the idea of treating nothing as a
thing.
This leaves us with the second choice,… simply change our intuitive idea of the notion of a set. A set is no longer considered to be a "collection of things". That is merely an
intuitive notion that is not needed for an axiomatic system to work. Instead, Cantor suggested, let's just ignore definitions, and forget about trying to comprehend the idea intuitively and make an axiom that simply states, "There exists an empty set". He somehow sold this idea to the mathematical community and they bought into it.
However, later on (possibly even after Cantor's death, I'm really not sure on that) the mathematical community started seeing other problems cropping up. For example, there must be a distinction between a set, and an element within a set. If this distinction isn't made then 1 would equal 0.
In other words, by Cantor's empty set theory
0 = {}
1={{}}
Now if there is no distinction between an element, and a set containing a single element then there is no difference between Cantor's definition of zero and One.
I'll grant you that this may appear quite trivial but I assure you that it is not.
The problem does not exist for Cantor's higher numbers because they are more complicated combinations of sets and elements.
2 = {{{}},{}} for example. Even if the elements were to stand alone there would be not be equal with any other number. In other words, you can remove the outermost braces (which merely convey the idea of a set) without reducing the contents (the actual collection of things) to configuration that represents a different number.
By the way this is much easier to see if you actually use the symbol err to represent the empty set. Or I think modern mathematicians use the Greek letter phi.
The bottom line to all of this is that Cantor's empty set theory is logically inconsistent. It's based on a logical contradiction of an idea of a collection of things that is not a collection of a thing.
So big deal you might say. It's a trivial thing just let it go and get on with using the axioms. Well if you think like than then you truly are a modern mathematician.
This logical contradiction does exists none the less, and it really does have an effect on the logic that follows from using these logically flawed axioms.
Two things should be apparent right way. First off, there is no formally comprehensible idea of a set. If you think of a set as a collection of things you are wrong. That is an incorrect idea in Cantor's set theory. It simply doesn't hold water in the case of the empty set which is the foundation of the whole theory.
There are far reaching logical consequences to this logical contradiction. And in a very real way they are almost like relativity. Just like relativity is hardly noticeable at small velocities, so the problem with Cantor's empty set theory is hardly noticeable for quantities much less than infinity.
But just as relativity comes into play and speeds that approach the speed of light, so do the logical errors of Cantor's empty set theory come into play at quantities that approach infinity, and more precisely it affects the concept of infinity immensely.
Georg Cantor is the only human ever to start with nothing and end up with more than everything. His set theory leads to ideas of infinities that are larger than infinity. In other words, it leads to the logical contradiction of some endless processes being more endless than others. If that's not a obvious logical contradiction I don’t' know what is, yet this absurd notion has been accepted and embraced by the mathematical community.
Finally, what I have typed into this post is merely the tip of the iceberg. I'm not about to write a book on an Internet forum to try to explain what most mathematicians should already be aware of. But there are other problems associated with the empty set theory as well, and they have to do with counting, or
countability.
The whole reason why Cantor introduced the idea of an empty set in the first place was to avoid having to tie the concept of number to the concept of the things that are being quantified. In other words, at the that time in history it was extremely important to the mathematical community that numbers be "pure". I might add that this was much more of a religious notion than a logical one so if we want to claim that someone was being religious rather than logical we should look at the mathematical community around the time that Georg Cantor lived!
In any case, by starting with nothing, Cantor was able to avoid having to deal with the concept of the individuality of the elements (or things) being collected. In other words, by starting with nothing he basically swept the prerequisite notion of the individuality of the objects that are being quantified under the carpet. And by doing so he has removed that constraint from any elements. In other words, in Cantor's set theory anything can be counted as an "individual" element even if it has no property of individuality. This is, in fact, the very reason why he is able to have infinities larger than infinity. He is actually counting objects that have no property of individuality yet he treats them as though they do. I can actually show the error of his ways using his diagonal proof that the irrational numbers are a larger set than the rationals say. Once you understand the whole problem concerning the property of the individuality of the elements it's a fairly obvious proof.
I don’t even know why I'm bothering to type this in actually. The mathematical community simply isn't ripe for this knowledge yet. It just isn't in the "air". I think that it will be soon though as more and more mathematician begin to study group theory where is most likely to become apparent. Some clever mathematician somewhere is bound to realize what's going on and become famous for discovering the "problem".
I think that the most important thing for mathematicians to realize is that this problem was only introduced into mathematics about 200 years ago. Compare that with the age of mathematics and we can basically say that it "just happened!".
I mean, correcting modern set theory will have no affect on things like Pythagorean's theorem, or Euclid's elements, or the vast bulk of mathematics including even calculus which came before Cantor's time. It's main impact with be in group theory. But group theory is becoming extremely important in modern science so it could end up having a big impact there.
Fortunately Cantor's illogical set theory won't affect most normal algebra or other mundane calculations, so don't expect to get tax refund checks from the government when the problem is finally corrected. The government never thinks in terms of sets anyway, they think in terms of bucks.
Science/philosophy decides what axioms you should be using, and mathematics tells you what those axioms imply.