No, no, no ......! You have challenged my axioms. The ball is in your court! :nerd:

"Infinity" in mathematics is just a name, a symbol that could be replaced with any other symbol — SophistiCat

More of a concept, actually. An expression "becomes infinite" if it grows without bound. No need for a symbol. But it's there to be used if one wishes.

Perhaps the rules or arithmetic are not so arbitrary. — Marchesk

Hmmm. Now where did we come up with a number system base 10?

The preference for consistency is one such rule. — Banno

Are my two axioms inconsistent? :roll:

Equivalence classes. Simple example: 1/3 = E = {n/m: n/m = 1/3}. Reflexive: shows 1/3 belongs to E. Symmetry: 1/3 = 2/6 and 2/6 = 1/3. Transitive: 1/3 = 2/6, 2/6 = 4/12 implies 1/3 = 4/12.

$X\ne X$

See Banno's Game.

The Axiom of No Choice: For any collection of non-empty sets there is at least one way to avoid choosing an element from each set.

(This will lead to a pathological nightmare in the case of an uncountable infinity of such sets)

The Axiom of Inclusion: Given two empty sets, one is the absence of an element of the other.

:nerd:

The sum of any two integers is zero.

Maths is constructed. One can do with it as one pleases with the symbols involved. We make the rules up as we go, and we can and do go back and change them as we like — Banno

It's a little more complicated than that. But go ahead and make one up. Should be fun. :smile:

Lots of changes coming, shifting agricultural patterns, beachfront properties inundated, and for those in the UK, get your woolies out when the Gulf Stream changes course. No turning back now. :sad:

All I said that what one could easily see even from this forum is that we do not understand infinity yet. — ssu

It's hard to argue with that. :chin:

Banach-Tarski paradox assumes that every object is the same size — Gregory

The Axiom of Choice sounds so very benign (one can pick an element out of each non-empty set in a collection of sets), but look what it leads to! And you think the Axiom of Infinity is bad!!! :scream:

"Complex analysis" (where complex means "having both real and imaginary parts") is hard to justify because we get our math from the world, where many is always many and not one. — Gregory

How about the real plane where a point is denoted by (x,y) ? Still have a problem?

Complex analysis may be hard to justify for those who know little mathematics, but ask a QM physicist about path integrals and their predictive values.

Here is a fundamental question: Is it reasonable (however defined) for philosophers who have not studied mathematics to argue basic principles of the subject? This is a far reaching question that can be applied to sciences in general. Remember, math came first, then its set theory foundations, PA and ZFC, were postulated in the 1800s and 1900s. Those mathematicians knew mathematics.

What about a "point" in N-dimensional space? There is no problem if a point is in essence an N-tuple of real numbers. Of course, for those who do not believe in irrational numbers there is no peace. :sad:

It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions. — Devans99

OK. Please describe those contradictions. I've not encountered them in my many years. :roll:

. . . you don't take into consideration at all that the now used axiomatic systems could be inconsistent. — ssu

All you have to do is come up with an example showing this to be the case, rather than argue in an abstract way about it. Maybe you have, as I haven't read all the posts. Good luck.

I see philosophy as a true science where we can gain 'true' knowledge. I'm not sure I could define the aforementioned quite well, myself. — fullofnull

As a math guy I might question "true science" , but I am not a philosopher so am biased. The only language games in which I indulge are crossword puzzles and they are not really games I suppose.

. . . the systems we do to keep ourselves busy and to keep ourselves distracted from death. — fullofnull

I hope this is tongue in cheek.

I think you will do fine on this forum, providing you stay away from the Axiom of Infinity.

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. — ssu

Wiki: incommensurable generally refers to things that are unlike and incompatible, sharing no common ground (as in the "incommensurable theories" of the first example sentence), or to things that are very disproportionate, often to the point of defying comparison ("incommensurable crimes"). Both words entered English in the 1500s and were originally used (as they still can be) for numbers that have or don't have a common divisor.

Not quite the same as mathematical measure theory. But the above may have more relevance to the thread.

Wiki: According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that a function is computable if and only if it has an algorithm.
Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet Σ = {0, 1}). The real numbers are uncountable so most real numbers are not computable. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant.

It's amazing at how well computers have served us, isn't it, given these restrictions? :cool:

With what?

Most working mathematicians do not feel mortally obliged to prove the existence of an object by producing the object. It is best if they can, however. If they can show that assuming the non-existence of the object infers a fundamental violation of mathematical reason - an indirect proof - that suffices. The Law of the Excluded Middle was quite useful in topology (sometimes referred to as math without numbers, with which I disagree).

I assume all of you grok the sophisticated presentations on this thread. No need to worry so much about all these technical details! Life will go on.

You guys are so HARD on Cantor! Are you not aware that normal, everyday mathematics produces excellent results? And much of that, done on a computer, treats all numbers as rational. Are you really that concerned with non-computable functions or non-measurable sets? Material like that in math is referred to as "pathological" frequently. :meh:

Very well put. :cheer:

The news media has become very selective in what it states and prints and is politically biased - on both ends of the political spectrum. How are philosophers to solve this problem?

Well, software, being of a modern generation of mathematicians, has opened my eyes again:

Wiki:"Other mathematical systems exist which include infinitesimals, including non-standard analysis and the surreal numbers. Smooth infinitesimal analysis is like non-standard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/ω, where ω is a von Neumann ordinal). However, smooth infinitesimal analysis differs from non-standard analysis in its use of nonclassical logic, and in lacking the transfer principle. Some theorems of standard and non-standard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. Statements in non-standard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis.

Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. Similarly, the Banach–Tarski paradox fails because a volume cannot be taken apart into points."

It IS possible to learn something on this forum! Thanks.

One-to-one and onto was not so difficult to say. But your generation learned differently. And it's the coin of the realm, now. :nerd:

Consider two infinite sets A and B of equal cardinality i.e. n(A) = n(B) = infinity
Shouldn't n(A) - n(B) = 0? — TheMadFool

Beyond finite instances, cardinalities are not numbers. They are equivalence classes.

Didn't Islamic science suffer from these doctrines? — Gregory

I don't know. But to some extent scientists - especially Greek - were sheltered by Muslim societies during the Dark Ages. Mathematical contributions are obvious, but while science was burned on the stakes by the Church Islam came to the rescue. But that was then.

I suppose religion is not PC, what with "Supreme Being" and all. It's a language minefield out there. :gasp:

The basic theory of probability and statistics are pretty well established, but when it comes to practical applications there are some problems. Remember when vitamin E was popular for one's health? Then another study showed it had a negative impact on health. Vitamin C is great for this and that, then suddenly it wasn't.

The medical profession does not have a sterling reputation for statistical studies. Sometimes this is due to multiple experiments in which outliers are given undue consideration. Sometimes the experiments are poorly designed.

However, having said this I will tell you that probability theory still leaves me a little uneasy, even though its applications have been largely very successful. Probability waves in QM? Who'd have thunk? :brow:

but wait, can a set be considered an element of itself??? :roll: — John Gill

Mathematical set theory and foundations provide a playground for combative metaphysicians. And if you contemplate my question you might see why.

Talk about thread-drift. I don't like the expression "quantum supremacy" because it implies no further scientific progress is likely in computing. Not because of PC, which I endured for years at the college level. How did this devolve into muscle-shirts? :brow:

Cardinality has to do with sets being in a 1:1 correspondence with each other. $\aleph_0$ is not a number in the common sense of the word. It denotes an equivalence class. It is not infinity. The terms "bijection", "injection", and "surjection" apply to functions and replace words such as "onto" or "into", which were easier to comprehend BITD, IMHO. They were introduced by a mysterious Frenchman, whose identity is baffling, in the 1930s. (Please don't try to explain "who" this was!) :nerd:

The proper subset of the infinite set itself has to be infinite — TheMadFool

Really? Consider N={1,2,3,...} and S={1,9} No 1:1 correspondence. You are missing "equivalence"

"A" proper subset? Not "The", I think.

Language frames much of our thinking, placing thoughts in a context that allows communication of those thoughts, sometimes imperfectly, to others. If not, why does this forum exist?

I'm probably hopelessly naive.

Axiom of Infinity is anything but established and self-evidently true. — ssu

And yet it sounds so simple. A set is a collection of "elements" or objects and the axiom says we can consider the collection itself as an object, but perhaps not the same kind of object - but wait, can a set be considered an element of itself??? :roll:

When you get down to the nitty gritty and use numbers, as in a computer program, they are all rational and thus countable. So how many rational numbers (fractions) do you think lie in the interval [0,1]? :chin:

I don't think measure theory has much to do with this. But maybe it does. A non-measurable set is not a triviality. A point on the real line simply corresponds to a specific real number. In the complex plane, a point corresponds to a specific complex number. If you don't accept Cantor's conclusions that the real numbers cannot be "listed" (counted) then you are destined to wander through a metaphysical jungle. Remember the suggestion: If you come to a fork in the road, take it. :cool:

It's interesting that the axiom of infinity and the axiom of choice can lead to such lengthy discussions. On the surface they seem so benign. The first simply says there is a set containing the natural numbers and the second says, roughly, that you can pick an item out of each set in a collection of sets. Most mathematicians go serenely about their investigations giving these two axioms little to no thought.

I have already given my opinions of infinities as being limit concepts, with virtually no mention of "the point at infinity". Infinitesimals, on the other hand, have been placed in a proper mathematical model and can be used to generate calculus. I would think these tiny little objects might generate more controversy than infinity, but apparently not. As part of the hyperreal number system they are intimately connected with infinities.

I never went beyond naive set theory, so these mathematical notions are merely mildly amusing. Keep in mind what I said before: the foundations of mathematics is a relatively recent subject that attempts to place all the math that has worked so well over millennia in "proper" logical frameworks. I suspect there are a few analytic philosophers on this forum who know far more than me.

www128.pair.com/r3d4k7/Mathematicae7.html

https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers


You can't even do links here it seems. This is a page on my website, and it comes up first time, then won't connect. It keeps returning to this forum. I don't know what is going on. The Wikipedia site keeps coming up. Not mine.

As a professional math guy, here is my opinion regarding infinity and the complex plane:

For me and my colleagues, |z| getting larger and larger without bound means z -> infinity. An actual point at infinity is irrelevant in practice. If I think of time going to infinity, I mean it in this sense. If you look at the projective plane sitting below the Riemann sphere, you can see z moving further and further out, without bound, and as it does so its projection on the sphere moves closer and closer to the north pole, but never reaches it.

OK. It requires $5/month to do so.

OK. Got the message. $5/month for permission to hot-link images. Maybe, maybe not. We'll see if it's worth it. :sad: