So that implies something false can be added to set theory without changing its consistency. — Devans99
I believe that the word "logical" is really used as a synonym of "consistent", but only in colloquial english. In mathematics I think it's not used at all.For example, if I define a simple maths system with only one number: 1 and one operator: + then I can axiomatically define 1+1=1. Its consistent but not logical. — Devans99
I feel that axiomising something that is clearly false in the real world (IMO) is the basis of my beef with set theory. — Devans99
But I would object to the idea of an actual completed infinite set existing in the real world. So a collection of objects whose number is not finite I would object to. — Devans99
I tried to google for "constructive real numbers are not complete", or something similar. — Mephist
I think this is what you refer to by "constructive reals". Is it?
Can you give me a link where is written that they are not complete? — Mephist
I am convinced that my definition of "constructivism" is not the same thing that your definition. — Mephist
Well, here's a simple definition of what I mean by constructive logic:
=== A logic is called constructive if every time that you write "exists t" it means that you can compute the value of t. === — Mephist
I believe that you can define real numbers that are complete in a constructive logic. I think the example that I gave you using Coq is one of these. But I could be wrong: I am not completely sure about this. — Mephist
I googled this: "non archimedean fields are not complete" and the first link that come out is this one:
https://math.stackexchange.com/questions/17687/example-of-a-complete-non-archimedean-ordered-field — Mephist
Probably, as they say, "The devil is in the detail". I read several times in the past about Abraham Robinson's hyperreal numbers, and I believe that I read somewhere that non archimedean fields are not complete. So I believe that, under appropriate assumptions, this is true. But why is this a problem? — Mephist
Hmmm... I understand what you mean:
- "constructive" reals are computable functions. Then there is a countable number of them — Mephist
- standard reals are the set of all convergent successions of rationals then their cardinality is aleph-1
- nonstandard reals are much more than this (not sure about cardinality), since for each standard real there is an entire real line of non-standard ones. — Mephist
Well, here's how I see it:
- "constructive" reals (with my definition) can be put in one-to-one correspondence with standard reals, only with a different representation — Mephist
(but I don't know a proof of this) and do not correspond to computable functions. It is true that if you can write "Exists x such that ... " then you can compute that x, But for the most part of real numbers x there is no corresponding formula to describe them (and this is exactly the same thing that happens for non constructive reals). — Mephist
- Robinson's nonstandard reals are more than the standard reals because you exclude induction principle as an axiom (so that "P(0)" and "P(n) -> P(n+1)" does not imply "forall n, P(n)"). But there are objects used in mathematics that are treated as if they were real numbers, but DO NOT have the right cardinality to be standard real numbers: for example the random variables used in statistics: — Mephist
https://en.wikipedia.org/wiki/Random_variable . So, they are more similar to nonstandard reals. — Mephist
- The real numbers of smooth infinitesimal analysis are less then standard real numbers, and even the set of functions from reals to reals is countable: basically, every function from reals to reals is continuous and expandable as a Fourier series. And there are infinitesimals.
What for such a strange thing? Well, for example, they correspond exactly to what is needed for the wave-functions and linear operators of quantum mechanics: there are as many functions as real numbers, and a real numbers correspond to experiments (then, there are a numerable quantity of "real" numbers). And what's more important, a wave function contains a definite quantity of information, that is preserved by the laws of quantum mechanics. — Mephist
So, from my point of view, there is not one "good" model of real numbers, at the same way as there is not one "good" model of geometric space. — Mephist
[ END OF PART TWO :-) ] — Mephist
Let's take as reference the most complete proof of the theorem that I was able to find: — Mephist
OK, this is the end. I think I cannot explain better than this my argument about BT. — Mephist
I think in that in an ideal mathematical language, Chaitin's Omega wouldn't be stateable. To say 'Omega is definable but non-computable' is surely not a statement about a number, but a statement about the syntactical inadequacy of our mathematical language for permitting the expression of Omega. — sime
There are two basic classes of infinite objects:
- Potential Infinities. The limit concept in calculus could be regarded as an example
- Actual Infinities. As represented by infinite sets / transfinite numbers in set theory — Devans99
Yes, and this clarifies a lot o things about infinity: — Mephist
"In first-order logic, only theories with a finite model can be categorical." (form https://en.wikipedia.org/wiki/Categorical_theory). ZFC is a first-order theory and it has no finite model (obviously), than it cannot be categorical. Ergo, you cannot use ZFC to decide the cardinality of real numbers: "if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities." (from the same page of wikipedia). — Mephist
Then, you can say, the problem is in the language: let's use a second or higher order language, and you can discover the "real" cardinality of real numbers. — Mephist
Well, in my opinion this is only a way of "hiding" the problem: it is true that if you assume the induction principle as part of the rules of logic, you get a limit on the cardinality of possible models (the induction principle quantifies over all propositions, so it's not expressible in first-order logic), but this is exactly the same thing as adding an axiom (in second order logic) and not assuming the induction principle as a rule. Ultimately, the problem is that the induction principle is not provable by using a finite (recursively computable) model: it's not "physically" provable. — Mephist
That's exactly the same situation as for the parallels postulate in euclidean geometry: you cannot prove it with a physical geometric construction (finite model), because it speaks about something that happens at the infinite, and the fact of being true or not depends on the physical model that you use: — Mephist
if computers that have an illimited amount of memory do not exist, or, equivalently, if infinite topological structures do not exist, then the induction principle is false, and infinitesimals are real! — Mephist
So, the sentence "if you use uncomputable (non constructive) axioms in logic, you can decide the cardinality of real numbers", for me it sounds like "if you use euclidean geometry, you can prove the parallel postulate". — Mephist
HOTT is not a constructivist theory (with my definition of constructivism) because it uses a non computable axiom: the univalence axiom — Mephist
This was considered by Voevodsky as the main "problem" of the theory, and there are currently several attempts to buid a constructive version of HOTT. One of them is cubical type theory (https://ncatlab.org/nlab/show/cubical+type+theory), but I don't know anything about it. — Mephist
[ THIS WAS THE LAST PART :-) ] — Mephist
Can you show me a physical theory, or a result of a physical theory, that is somehow derived from the fact that a continuous line is made of an uncountable set of points? — Mephist
Formal logic (currently assumed as the foundation of mathematics) is only dependent on one very fundamental fact of physics (that usually is not regarded as physics at all): the fact that it's possible to build experiments that give the same result every time they are performed with the same initial conditions. — Mephist
Mathematics (what is called mathematics today) is the research of "models' factorizations" that are able to compress the information content of other models (physical or purely logical ones). A formal proof makes only use of the computational (or topological) part of the model. The part that remains not expressed in formal logic is usually expressed in words, and is often related to less fundamental parts of physics, such as, for example, the geometry of space. — Mephist
Riemann understood that the concepts of "straight line", measure, and the topological structure of space are not derivable from logic, but should be considered as parts of physics. — Mephist
In te future, when mathematicians will start to use quantum computers to perform calculations, — Mephist
I believe that even the existence of repeatable experiments will not be considered "a priori", but as an even more fundamental part of physics. — Mephist
So, there will be quantum logic that is more powerful than standard (or even constructionistic) logic, at the price of not being able to be 100% sure that a proof is correct (but you will be able, for example, to say that we are sure about this theorem with 99% of probability). — Mephist
Surely your ( and most of other peoples' ) reply to what I just said is that "this is no more mathematics". — Mephist
Well, at the time of Euler topology was not mathematics either. — Mephist
There's exactly one model (up to isomorphism) that is Cauchy complete; and that is the standard reals. — fishfry
That seems to be the common point of all your arguments about real numbers, so I wanted you to show you this: — Mephist
Here is a good explanation of what "contructive mathematic" means: https://www.iep.utm.edu/con-math/ — Mephist
Would 1/0 = a vacuum/blackhole? -abstract thoughts — Pomme
Assuming this is correct, which given your penchant for error and misstatement is going some. But given this, let's just consider the stronger which I assume will carry the lesser, actual infinite sets. If there are actual infinite sets then it would seem to follow a fortiori that there wold be potentially infinite sets.
Question: how many numbers are there? It's a fair question and a meaningful question. Answer!
Does the answer mean that you could produce a bean for every number and thus have a very large pile of beans that extended out of sight? Of course not! But the concept for the purpose is complete.and actual. QED. — tim wood
So actual infinity is not realisable in our minds or in reality — Devans99
Are we the standard by which we measure infinity? Isn't this like saying tortoises that live upto 300 years don't exist because humans only live upto 100 years? — TheMadFool
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