What about fractals? These are infinite spaces that you can explore on a computer. Their instantiation in a computer realizes what was an abstract infinity. — hypericin

Fractals are generated by simply iterating certain complex functions. They are not "infinite spaces" but images on computer screens. The iteration process is finite, say n=1000. There is no abstract infinity other than implied replications of patterns like turtles all the way down. You never really get there.

What is a potential infinitesimal — jgill

It is the reciprocal of a potentially infinite number, e.g. a random value taken from the codomain of the rational valued function 1/x.


'non-standard analysis' is the correct umbrella term, but it is already befuddled by the various alternatives that fall under it, some of which receive rightful criticism for obscuring matters even further, e.g the hyperreals .

You mean without absolutely infinite sets, presumably. — sime

Why in the world would I couch anything in terms of "absolutely infinite"? The notion of "absolute infinity" does not occur in classical mathematics since Cantor was superseded by Z set theory, as I had already alluded to:

Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory? — TonesInDeepFreeze

And you skipped that question. Would you please answer it?

You keep talking about "absolute infinity" but in absence of any reference to it in mathematics after Cantor, and in absence even of your own definition of it.

approximating an ideal, abstract and causally inert mathematical derivative. — sime

Mathematical objects are in general abstract. And knowing what you mean by "causally inert" awaits knowing your definition.

[a potential infinitesimal] is the reciprocal of a potentially infinite number — sime

A philosophical view, such that we are already familiar with, on the notion of 'potential infinity' is of course welcomed. And I have little doubt that it has been formalized somewhere. But you again skipped my point that it is only by seeing a particular formulization that we can compare its ease of use with classical mathematics. From the (admittedly not deep) reading I have done, generally alternatives to classical mathematics are much more complicated to formulate.

Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one).

Then, aside from the mathematical and philosophical subject, we now are talking about heuristics in the form of pedagogy. Nothing you have mentioned so far appears to be any more pedagogically promising than the classical definition of 'the derivative'. Indeed, there is no reason to think that a high school or college student would not have a much harder time grasping non-standard analysis combined some way with "potential infinity" than grasping the usual notion of a derivative. Nor have you pointed out any instance in which scientific or engineering calculations would be better enabled by your seemingly personal graft of non-standard analysis with a notion of 'potential infinity'.

obscuring matters even further, e.g the hyperreals . — sime

Since hyperreals are formalizable in set theory, they are formalizable in internal set theory too.

You simply skipped the point that mathematical definitions are never circular or tautologies. — TonesInDeepFreeze

You skipped it again.

That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".

Moreover, most remarkably, in another thread:

https://thephilosophyforum.com/discussion/comment/653236

you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that. — TonesInDeepFreeze

You skipped that point twice now. Maybe you'll address it later, but as it stands, you fail to substantiate your claim as to what Brouwer or any intuitionist or any constructivist or any mathematician or philosopher ever said.

Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one). — TonesInDeepFreeze

Overall, good observation.

To my knowledge, it isn't possible to point to a complete formalisation of potentially infinite logic, because it doesn't yet exist. All we have are fragments or incomplete axiomatizations of the concept, that have been invented by different logicians over the years with respect to different systems for purposes other than the current discussion. This is somewhat similar to the proliferation of different programming languages. There isn't any inter-subjective agreement as to how to formulate open-world reasoning.

- The I, S and T axioms that Edward Nelson introduced are useful for formulating what it means to reason with as-of-yet unconstructed elements of an unfinished set, in spite of the fact he proposed the axioms in the context of ZFC as an alternative to model-theoretic non-standard analysis in ZFC. ZFC is of course inadmissible for the purposes of this discussion, and is the reason why his formalisation doesn't tend to be associated with formalizing potential infinity. But the I,S and T axioms, divorced from the problematic axioms of ZFC appear to be a relevant fragment of some formalization of potentially infinite logic.

-Brouwer's notion of choice sequences, i.e. unfinished sequences, serve as the template for potentially infinite sequences but his formulation doesn't to my understanding provide what I,S and T does.

Choice sequences allow the expression of unfinished sequences, e.g

{1,2,3,...}, where the dots "..." are understood to mean "to be continued"

Brouwer introduces continuity axioms that define what it means to prove a universal proposition over a domain that consists of such unfinished sequences. However, his concepts, at least to my understanding, doesn't permit direct talk about numbers that we have presently declared, but cannot currently quantify, e.g. "The height of the tallest human being who will ever live"

- Linear Logic, as opposed to intuitionistic logic, is the logic i would associate with Intuitionism and potential infinity, because it is a resource conscious logic. Again, as you might say, it is "not evidently associated with p.i", especially in view of it's exponential fragment. Squinting at axioms to see their practical significance is still an unfortunate necessity.

ZFC is of course inadmissible for the purposes of this discussion — sime

For potential infinity. Though, in ZFC, we can describe sequences of finite sets.

the I,S and T axioms, divorced from the problematic axioms of ZFC — sime

Problematic for some people, not for others. And I am pretty sure that if you take ZFC out from under IST you're left with a theory that accomplishes quite little.

appear to be a relevant fragment of some formalization of potentially infinite logic. — sime

Has anyone tried?

Linear Logic — sime

It looks very interesting. But you couldn't seriously propose it as a way for college freshmen to learn calculus.

Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer. And I still don't know why in the world you go on about "absolute infinity", which is a notion pretty much entirely unused since Cantor was superseded by axiomatic set theory.



.

Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer — TonesInDeepFreeze

Nope. You need to reread the article.

There are no passages in the article that support your misrepresentations. If you think there are passages that support you, then you can cite them specifically.

Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".

You fabricated those.

And what was your point arguing about "absolute infinity", which is a notion that has virtually been out of play since axiomatic set theory? For that matter, as I've already asked, what is your definition of "absolute infinity"?

For that matter, as I've already asked, what is your definition of "absolute infinity"? — TonesInDeepFreeze

:up:

For that matter, as I've already asked, what is your definition of "absolute infinity"? — TonesInDeepFreeze

Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity, relative to which the existence of said entity cannot be independently verified, empirically evaluated, or constructed with respect to a finite amount of data.

Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. Quine's famous example "All Bachelors are unmarried men" can be held as being true by definition in the mind of a particular speaker, but in doing so it can no longer be regarded as being representative of how a community of speakers might use the words "bachelor" and "unmarried man", given the limitless potential of them using the words non-equivalently.

Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension, whereupon it is inconsistently alleged that a finite description of a function can somehow represent a limitless amount of information that is also exact.

The alternative interpretation corresponding to "potential infinity" is to consider the definition of such entities as being vague and verifiable, as opposed to being semantically precise but unverifiable.

Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded". — TonesInDeepFreeze

It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals.

Actually my phrasing is a slightly weaker statement, considering that potential infinity is usually used in the context of monotonic sequences, as in infinitesimals or infinitely large numbers. The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. This constitutes a random stopping event, in the sense that the time of the pause is not defined a priori at the time when dx is declared to be infinitely small or x to be infinitely large.

Defining potential infinity in terms of a "non-terminating" process is problematic however, given the fact that 'non-termination' isn't a verifiable proposition if interpreted literally, which is a concept belonging to absolute infinity. What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. Then consider the fact that any of the finite extensions generated by pausing a process are countable and isomorphic to the integers. These are my considerations when thinking of "potential infinity" in terms of a priori unbounded finite numbers instead of in terms of a "non-terminating" process. However, my definition might cause confusion due the fact it is more general and includes random variables with unbounded values.

Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension . . . — sime

Apart from Cantor's speculations I never came across this idea. It can't be applied to sets so it sits out there in an unattainable splendor, ignored by most in the profession.

It would be nice if you addressed my points directly aside from (or at least in addition to) whatever other meandering about various subjects you have in mind.



Instead of defining 'absolute infinity' you give various notes on your ideas about it.

'absolute infinity' is a noun. For mathematics, the definition of a noun requires the definiendum (a noun phrase for defining a noun) on the left and a definiens (a noun phrase for defining a noun) on the right.

So a definition of 'absolute infinity' would be of this form:

absolute infinity = [insert a mathematical noun phrase here]

/

in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension — sime

There you still don't provide a definition, but you take 'absolute infinity' to refer to a certain kind of "interpretation". Here are the problems for you:

(1) You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it.

Accepting that there is a common notion of an intensional definition as opposed to an extensional definition, we still see that 'an extensional definition' just boils down to the definiens being a finite list. But a set is infinite iff it has no finite listing. So there is nothing gained from saying 'absolutely infinite' rather than merely saying 'infinite'. The definitions:

x is finite iff x is 1-1 with a natural number*

x is infinite iff x is not finite

Then your sense of 'absolutely infinite' is just 'infinite'.

Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity — sime

Then syntactic infinitistic mathematics is not itself liable.

Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. — sime

What example do you offer of a mathematician or philosopher of mathematics making that mistake?

total function — sime

What total function do you offer as an example? 'total' is better thought of as a 2-place relation. A function is total on a set iff the domain of the function is that set. So what exactly do you have in mind regarding totality? And, for example, as to the the infinitude of the set of natural numbers, what total function do you think is improperly employed?

/

Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
— TonesInDeepFreeze

It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals. — sime

No it is not "practically equivalent". You ADDED "finite", "a priori indefinite", "finite entities" and especially "eventually finitely bounded". Those are not mentioned, neither literally or practically, in the article's explanation of the notion of potential infinity (except 'finite' is mentioned regarding the finitude of each natural number). Generally, the article does not say that potentially infinite sequences are finite sequences. Even for an intuitionist, a finite sequence is not a potentially infinite sequence, notwithstanding that only a finite portion of a potentially infinite sequence is constructed at any given point. To really stress the point, especially, the article says nothing like "eventually finitely bounded".

The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. — sime

Please cite a source where that is mentioned as part of the "common definition".

when dx is declared to be infinitely small or x to be infinitely large. — sime

You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off from them.

/

* That is circular with my definition of 'is a natural number' in another thread, but that can be overcome by using a different definition of 'is a natural number' such as 'an ordinal that is well ordered both by membership and the inverse of membership').

Simply:

Where do you find "eventually finitely bounded" in Brower, or even any secondary source, on potential infinity? Please cite a specific passage.

Again, I'm not asking about your own notions; I'm asking where you got your idea as to what Brower said about it.

And you might hold off snidely telling me to "reread the article(s)" when you still don't cite specific passages in an article and the article you mentioned does not at all support your claim.

What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. — sime

I think you may have misinterpreted the Wikipedia article:

potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps

There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition.

You have a very sophisticated writing style that I find a bit hard to process, but others may not. And when a person writes well there is a temptation to put aside statements that might give one pause. I'm guilty, but fortunately TonesInDeepFreeze is sharper in this regard.

There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition. — jgill

If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?

I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.

The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future.

Simply:

Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage. — TonesInDeepFreeze

It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject, and I have already presented my arguments in enough detail as to why it is better to think of potential infinity in that way.
.

You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off fro them. — TonesInDeepFreeze

.


Intuitionism is partially aligned with constructively acceptable versions of non-standard analysis. If you want an more authoritative but easy-read sketch, Read Martin Lof's "The Mathematics of Infinity" to see the influence Choice sequences have had on non standard extensions of type-theory (which still cannot fully characterise potential infinity due to relying exclusively on inductive, i.e. well-founded types.

You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it. — TonesInDeepFreeze

Classical mathematics and Set theory conflate the notions of absolute with potential infinity, hence only the term "infinity" is required there. Not so in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk.

Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory:

1) The Law of Excluded Middle is not only invalid, but false with respect to intuitionism.

2) The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity.

For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..."

3) Choice Axioms obscure the distinction between intension and extension, whereupon no honest mathematician knows what is being asserted beyond fiat syntax when confronted with an unbounded quantifier.

3) The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. Not so according to potential infinity, since it cannot be determined that two functions are the same given a potentially infinite amount of data .

We also have Markov's Principle: according to absolute infinity, an infinite binary process S must contain a 1 if it is contradictory that S is constantly zero, and hence MP is accepted. Not so according to potential infinity, due to the fact that 1 might never be realised. This principle is especially relevant with respect to Proof theory, since any proof by refutation must eventually terminate at some point, before knowing for certain whether an unrefuted statement is refutable. So unless we are a platonist who accepts absolute infinity, Markov's principle isn't admissible.

If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?

I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.

The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future. — sime

We seem to be looking at a common notion from different perspectives. I assume you are a CS person (?). First of all a student in a math class is more likely to ask, What is an infinite sequence?, whereupon the definition as a function of the positive integers is provided. Then comes the distinction between a sequence in which each term is defined by a formula and one that self-generates by recursion. Examples would make these ideas clear:

(1) ${{S}_{n}}=S(n)=3n+1,\text{ }n\in N$

(2) ${{S}_{n}}={{S}_{n-1}}+{{S}_{n-2}},\text{ }{{S}_{1}}=1,{{S}_{2}}=1,\text{ }n\gt2$

Then the idea of a finite expansion (largest value of n) or an infinite expansion (non-terminating).


$\lt{{S}_{1}},{{S}_{2}},\cdots ,{{S}_{n}}\gt\text{ , }\lt{{S}_{1}},{{S}_{2}},\cdots \gt$


Then, of course, there are special and/or more complicated cases, like a sequence which is non-terminating but assumes the same value after a certain point, or a sequence that is convergent in a metric space, etc.

In all my years I don't recall using the expression "potential infinity".

If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity? — sime

Look in virtually any introductory textbook or set of lecture notes on computability. And we don't need a notion of 'potential infinity' to explain the notion of non-termination. The classical treatment of computability is replete with the notion of non-halting. For example, it is a well known simple fact that a program to list the natural numbers does not halt.

Moreover, it is the responsibility of proponents of the notion of 'potential infinity' to provide the needed definitions to support the notion; not the responsibility of people who don't rely on the notion. The fact that you are mixed up about this subject shouldn't entail that you try to patch that up by supplying incorrect and incoherent claims and attribute them to Brouwer and the intuitionists. You continue to say that Brouwer and intuitionists understand potentially infinite sequences to be "eventually finitely bounded". Yet, after multiple requests, you fail to provide a source where Brouwer or anyone said that.

I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. — sime

Classically, it can be explained as not one finite sequence but as a sequence of finite sequences. Or, non-classically, to avoid having an infinite sequence of finite sequences, as anyone can read in virtually any article about 'potential infinity', even at the most basic level: For any finite sequence, there is a finite sequence of greater length, but there is not an upper bound to the lengths of such finite sequences. That is the OPPOSITE of saying that there is a finite bound on the lengths.

"non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc. — sime

One can posit that no physical process continues without termination. But, as I've asked you again and again, please cite where Brouwer said that a potentially infinite sequence is "eventually finitely bounded". More generally, if you have no Brouwer source to point to, then you should not conflate your claim that an ideal process is not realized physically so a non-terminating process is "eventually finitely bounded" with Brouwer who, as far as we know, never advocated that a potentially infinite sequence is "eventually finitely bounded", especially as the notion of potential infinity is the OPPOSITE.

Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage.
— TonesInDeepFreeze

It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject — sime

You use terminology in such a sloppy yet grandiose way:

(1) Logical equivalence is a special notion. You haven't shown any "logical equivalence".

(2) By mere fiat you declare a logical equivalence. Still, you do not cite anything Brouwer said that even suggests (let alone is "logically equivalent") to "eventually finitely bounded", especially as it is the OPPOSITE of the notion of potential infinity.

Martin[-]Lof's "The Mathematics of Infinity" — sime

I will look at it. I am not well versed in type theory and category theory.

Classical mathematics and Set theory conflate the notions of absolute with potential infinity, — sime

ZFC could not possibly conflate the notions since set theory doesn't even have a notion of 'potential infinity' nor does set theory mention 'absolute infinity'.

Again, 'absolute infinity' is a notion of Cantor that is not used in ZFC. You persist to use 'absolute infinity' in your own personal sense (for you, 'absolute infinity' is the notion that an intensional definition can specify a set that cannot be finitely listed), which is very different from Cantor's use of the term.

And even with your sense, though ZFC does define sets that are not finite, that is not "conflating" with some other notion ('potential infinity') that ZFC does not even address.

in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk. — sime

A rigorous definition would be good for those who use the concept. But ZFC does not use it. And ZFC does provide a rigorous axiomatization for classical computer theory. If you want a rigorous non-classical computer theory, then it's your job to make it rigorous; your lack of doing that is not a fault of ZFC.

And I guess there are some people in computer science who regard ZFC as junk, but that is not at all any kind of consensus or, as far as I know, even a large contingent. You don't legitimately get to speak on behalf of "computer science".

Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory: — sime

Wow. You are so wrong, and so obviously so. A combination perhaps of ignorance and intellectual dishonesty.

The very purpose of Z is to not include axioms that would allow an absolute infinity. Why don't you get a book on introductory set theory and inform yourself on this subject?

The Law of Excluded Middle is not only invalid, but false with respect to intuitionism. — sime

Yes, LEM is not a logical validity in intuitionism. That doesn't contribute to claiming that Cantor's sense of 'absolute infinity' is the same as your own personal sense of it. (Also, there is a technical question about what 'false' in a semantic sense means for intuitionism. In finite domains, LEM is TRUE in intuitionism. Of course, we would have to look at specific intuitionistic set theories to see whether LEM is false in any given model.)

The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity. — sime

I have no idea what you have in mind with the notion of the axiom of regularity "preventing unfinishable sets". ZFC has no predicate "unfinishable sets", so the axiom of regularity couldn't allow nor "prevent" anything about. It is incorrect to say that ZFC makes determinations on notions that are not even expressed in ZFC.

And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.

For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..." — sime

I don't know what that is supposed to mean. But, to be clear "..." is not in the language of set theory, not even as extended by definition, but rather it is informal notation that can be eliminated with actually rigorous notation.

no honest mathematician knows what is being asserted [with the axiom of choice] beyond fiat syntax when confronted with an unbounded quantifier. — sime

The axiom of choice is intuitive. One is free to reject it, but you are incorrect to say that mathematicians don't know what the axiom asserts.

And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.

The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. — sime

No, according to what YOU mean by 'absolute infinity'; it's not what Cantor meant.

And the axiom of extensionality doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.

Even more basically, even though ZFC does capture a great deal of Cantorian set theory (but not including Cantor's 'absolute infinity'), that does not entail that what Cantor meant by 'absolute infinity' is what you mean by it.

Condensed:

1. sime's claim of what Brouwer meant is the OPPOSITE of the basic notion of potential infinity, and sime has shown no source that supports his claim.

2. sime is plainly incorrect that Cantor's notion of absolute infinity is the same as sime's, and sime's argument about ZFC in this context is incorrect since ZFC dramatically DIFFERS with Cantor on the matter.

:broken:

From the mole in chemistry and nuclear physics to the relative nature of time the universe has yet to yield a single finite limit for anything outside of human measurements.

We can measure things and we can make linear statements about those things with mathematics but beyond that we have to use linear algebra. There is a working axiom in mathematics (not yet extremely popular but growing every year) that linear algebra should be the core axiom of mathematics.

Traditionally the first axiom of mathematics is commutative principle. Increasingly, commutative principle is seen as the first derived axiom with the first being the axiom that given any observable set we can assign whole numbers to that set as a form of measurement.

Take it how you will, the simple truth is that it's the psychology of individuals and their personality cults that have had the most influence on how everyday people understand the universe, not the ideas of finite and infinite. Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything.

We can measure things and we can make linear statements about those things with mathematics but beyond that we have to use linear algebra — SkyLeach

Lots of non-linear material out there. That's where the study of differential equations gets interesting. So much more.

1 apple + 1 apple = 2 apples
1 infinity + 1 infinity = 1 infinity :chin:

Infinity broke math. Leopold Kronecker 1, Georg Cantor 0. Sorry Cantor old chap, I'm a big fan, but Kronecker's got a point!

Infinity broke math — Agent Smith

Necessary to allow the goodness to pour forth. :cool:

Y'know that's not actually a true statement. You can't have equality without proof and ♾+1>♾
so ♾+1!=♾

All it takes is shifting the axioms in the philosophy of mathematics down by one. Seems reasonable to me.

Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything. — SkyLeach

It's interesting you have come upon a well-kept secret within the mathematics community. I, myself, have witnessed such rants at the top echelons of the profession. It's an embarrassing spectacle and rather unexpected within a discipline so rigorous and demanding calm reasoning. Sad. :cry:

From a psychology (especially evolutionary and developmental) perspective it does make a fair amount of sense.

People gifted in mathematics tend to be (very much a generalization) very judgemental, love symmetry, almost obsessively orderly, emotionally distant, become easily obsessed with problems, stoic in their self image, etc...

Of course there are often exceptions to this as a person can be a brilliant mathematician but not see themselves as naturally gifted. Einstein comes to mind as he forced himself to do math in order to explain his theories but preferred to visualize stuff in his head and then labor to write it down as equations.

Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be.

One of my favorite examples of the difference between linear calculus and set theory elegance is to compare Euler Angles with Quaternions. In linear algebra the quaternion equation iterates over vectors and translates (rotates) their position in 3 coordinate planes. It tells you the new location of each member of a vector. With Euler angles, however, the description of the matrix is what gets rotated, not the matrix.

Not sure where set theory comes into play here. Quaternions are more algebraic and geometric. But I will admit the part of calculus I least enjoyed teaching is spacial coordinates and rotations and translations. And the closest I've come to exploring Euler angles and quaternions is in SU(2):Dynamics of LFTs of SU(2)

Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be — SkyLeach

I can write very concise and precise definitions and descriptions, then use them in sheer babble.

There is a working axiom in mathematics (not yet extremely popular but growing every year) that linear algebra should be the core axiom of mathematics. — SkyLeach

I'm not familiar with that. Where can I read about it?

Traditionally the first axiom of mathematics is commutative principle. — SkyLeach

I've never seen such a "first axiom of mathematics" as you describe it. Is it something you've read or just an idea of your own?

derived axiom — SkyLeach

What is a "derived axiom"? Ordinarily, sentences derived from axioms are called 'theorems'.

the axiom that given any observable set we can assign whole numbers to that set as a form of measurement. — SkyLeach

Where is that stated as an "axiom"?

Hand them ["famous and influential mathematicians"] a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything. — SkyLeach

What famous mathematicians in particular do you have in mind? Where have you witnessed these "meltdowns"?

The use of sets is ubiquitous in mathematics. I didn't know that famous mathematicians were having "meltdowns" about sets.

1 infinity + 1 infinity = 1 infinity — Agent Smith

Again, there is no object that is infinity (other than such things as points of infinity on extended numbrer lines). There are sets that have the property of being infinite.

That cardinal arithmetic is idempotent for infinite sets (especially for the set of natural numbers) is not really not problematic if you bother to read the proof.