Personally, I don't take the first one as involving the extended reals.

I take 'inf' in

lim[x -> inf] 1/x = 0

as notation that unpacks as:

0 = the limit of the function 1/x where x ranges over the positive natural numbers

So there is no need to involve an object named 'inf'. It's a more parsimonious approach.

I'm not sure offhand how I would unpack lim[x -> 0] 1/x = inf; but my hunch is that I could do it.

Limiting ourselves for the sake of convenience to just Achilles and the tortoise, what exactly is there to debate about? — tim wood

What should be the payoff if you bet 1000 euros on Achilles.

So there is no need to involve an object named 'inf'. It's a more parsimonious approach. — TonesInDeepFreeze

I did say that the extended reals are essentially a notational convenience. Technically we could live without them. But why? By that reasoning we should call googolplex SSSSSSS...SSSS0 with the appropriate number of successors. But isn't $10^{10^{100}}$ better?

Notation is all about convenience of expression.

After all, when you write, "0 = the limit of the function 1/x where x ranges over the positive natural numbers," what do you mean by limit? Do you mean that for all epsilon there exists a delta such that ... etc? Where does your definitional parsimony end? If I say, "Let X be a topological space," should I replace that with, "Let X be a set along with a collection of its subsets satisfying ..." and then repeat the definition of a topology? That would get tedious fast.

I don't see a need for disagreement here. You can notate as you wish; and I can say why I also use that notation but like to point out that 'inf' is dispensable when we unpack, which has a pedagogical purpose: Many people, such as in Internet threads, are clueless about axiomatic, rigorous mathematics, so they have the misconception that 'inf' must name a certain entity. So it is instructive to explain that it is merely contextual notation that does not invoke any entities other than real numbers and a function.

I don't see a need for disagreement here. — TonesInDeepFreeze

I don't either. I agreed with you. I originally said that the extended reals are essentially a notational convenience. As are the names of the numbers in decimal notation, as I indicated.

I agreed with your point. But that doesn't mean we shouldn't use notational conveniences.

You can notate as you wish; and I can say why I also use that notation but like to point out that 'inf' is dispensable when we unpack, which has a pedagogical purpose: Many people, such as in Internet threads, are clueless about axiomatic, rigorous mathematics, so they stubbornly assert that the notation with 'inf' must mean that it names a certain entity. So it is instructive to explain that it is merely contextual notation that does not invoke any entities other than real numbers and a function. — TonesInDeepFreeze

It does name a certain entity; namely the larger of the two extra points in the extended real numbers. I hope you'll give that Wiki page a read, it's not bad as Wiki pages go.

The system of extended reals is rigorous. And we can define '-inf' and '+inf' in a way to instantiate the system. Then, one may wish to define such notation as 'lim[n -> inf] f(n)' where 'inf' does denote and is not merely a contextual notation. That's all fine.

But there's another approach, in which we don't need to have the system of extended reals for most of ordinary analysis, but rather we take the notations '-inf' and '+inf' as not denoting but rather as merely contextual.

And we can use either approach as suits the discourse.

Achilles and the tortoise combine the "fact" (?) of infinite divisibility of space with the arrow paradox. The latter doesn't seem to understand propulsion and inertia but the former has always confused me. I would expect there to be something discrete and yet still space at the bottom of divisibility, but that seems like a contradiction in terms. Discrete can only mean points (it would seem). And sure, you can say that you imagine an infinity of them in a finite space, but you cant really imagine anything infinite. Only reason can understand the infinite, not imagination (it would seem). So how does the principle of infinity unite with the principle of finitude in order to have a geometrical object? To put it philosophically..

Yes exactly. — fishfry

Thanks for the clarification. Can we calculate the contribution of the $\omega$th term to the series?

Yes and thanks.

Can we calculate the contribution of the w-th term to the series? — MoK

What series? A series is an infinite summation, which is the limit of a sequence of finite sums; ordinarily the domain is w, so there is no w-th involved; w is not a member of w, so w is not in the domain of a sequence whose domain is w. There are sequences in which w is in the domain (such as the domain being wu{w}), but I don't know of one that is a series (maybe there is such a thing?).

Only reason can understand the infinite, not imagination (it would seem). — Gregory

That seems well put.

I use reason to formally conceive that there are infinite sets. It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle).

But I can't informally see in my "mind's eye" an infinite set. Fortunately, though, I don't study mathematics with a dictate that I must reject formal, deductive understandings merely on the basis that my mind's eye doesn't visualize them.

One problem though does haunt me: Every arithmetical statement is either true or false. There is a function that determines the truth or falsehood of every arithmetical statement. But, of course, it's not a computable function. The truth or falsehood of every arithmetical statement is determined, but there are arithmetical statements of which we could never find the determination. It's as if those statements and their determinations are "out there floating around" but I can't visualize what it means that they are true or false except that I know there is a function that determines them.

What should be the payoff if you bet 1000 euros on Achilles. — TonesInDeepFreeze

Not much, I should think. I wonder what odds Zeno might give.

..

That was Zeno's scam. He conned people into thinking that Tortoise had just as good a chance as Achilles, then he took people's bets on Tortoise. It's talked about in the books "Conundrums For Dummies" and "The Complete Idiot's Guide to Greek Antinomies".

Thanks for the clarification. Can we calculate the contribution of the ω
�
th term to the series? — MoK

It's not any term of the sequence. It's the LIMIT of the sequence. I know you took calculus a long time ago. They taught you that 0 is the LIMIT of the sequence 1/2, 1/4, 1/8, 1/16, ..., correct?

0 is not ANY TERM of the sequence. It's the LIMIT of the sequence.

I use reason to formally conceive that there are infinite sets. It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle). — TonesInDeepFreeze

Reason is utterly insufficient to determine if there are infinite sets.

The only thing that guarantees the existence of an infinite set is the axiom of infinity.

If we adopt the axiom of infinity, there are infinite sets. If we reject the axiom of infinity, there are no infinite sets. It's as simple as that. The existence of an infinite set is purely a matter of accepting or rejecting the assumption that says there is an infinite set.

Therefore reason can not possibly determine the matter. It comes down to a personal choice, since both the affirmation and negation of the axiom of infinity are consistent with the other standard axioms.

It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle). — TonesInDeepFreeze

Your principle leads directly to a contradiction. The restriction needed to patch the problem is restricted comprehension, which would already require there to be an infinite set that's a superset of the infinite set you wish to conjure. Without the axiom of infinity, THERE IS NO SET containing all the natural numbers.

One problem though does haunt me: Every arithmetical statement is either true or false. — TonesInDeepFreeze

No. Just no. Given an arithmetical statement (a syntactic entity) AND AN INTERPRETATION of the symbols of that statement (semantics) then the statement becomes true or false.

Example: Given X and Y, there is Z such that Y + Z = X.

This is TRUE in the integers; and FALSE in the natural numbers.

So you are wrong that every arithmetical statement is either true or false. Absent an interpretation, a statement has no truth value at all.

Reason is utterly insufficient to determine if there are infinite sets.

The only thing that guarantees the existence of an infinite set is the axiom of infinity. — fishfry

I should not have said 'formally'. I meant informal deduction. My point was to contrast informal deduction with "mind's eye" visualization. I reason from the notion that there is the property of being a natural number. And I don't claim that reasoning is as rigorous in avoiding paradox as axioms. My point was at a very broad level of generality, not necessarily responsible to formal contradictions. I should have counted on someone not recognizing that my remarks are merely in the spirit of personal reflection. Moreover, I am not advocating such reflections as philosophical conclusions that others necessarily should adopt.

If we adopt the axiom of infinity, there are infinite sets. If we reject the axiom of infinity, there are no infinite sets. It's as simple as that. — fishfry

It's not as a simple as that, and it is not correct:

(1) To reject the axiom of infinity is to not include it in the axioms. Not including the axiom of infinity does not entail that there are no infinite sets. Rather, to entail that there are no infinite sets requires both not including the axiom of infinity and also adopting the negation of the axiom of infinity.

(2) The axiom of infinity is used for many axiomatizations. It is not precluded that one may axiomatize differently and still derive the theorem that there exist infinite sets. It is fine to say that with ZFC, for example that the axiom of infinity is required to prove the existence of an infinite set; but my remarks were not specific to axiom systems.

The existence of an infinite set is purely a matter of accepting or rejecting the assumption that says there is an infinite set. — fishfry

Again, I should not have said 'formal'. I meant informal deduction.

It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle).
— TonesInDeepFreeze

Your principle leads directly to a contradiction. The restriction needed to patch the problem is restricted comprehension, which would already require there to be an infinite set that's a superset of the infinite set you wish to conjure. — fishfry

(1) The axiom of infinity provides that there is a successor-inductive set. The axiom of separation provides deriving the set of natural numbers from the fact that there exists a successor-inductive set. w is the unique set that is a subset of all successor-inductive sets. There is no particular proper superset mentioned. Only that there is a successor-inductive set (whether or not a proper superset of any other successor-inductive sets). Then, after we have proven that there is a unique set that is a subset (notice: not even presuming that it is a proper subset) of all successor-inductive sets, we prove that there are proper supersets of w that are successor-inductive (e.g. w+w).

(2) I only described (subjective) reasoning, not even associated with any particular axioms. I mentioned that the principle "for every property there is a set" would not be taken without restriction (if I were to expand on that remark, it would be that such properties as "is not a member of itself" or more generally, unrestricted comprehension don't fly). Taking the property "is a natural number" to provide the set of natural numbers is not contradictory.

One problem though does haunt me: Every arithmetical statement is either true or false.
— TonesInDeepFreeze

No. Just no. Given an arithmetical statement (a syntactic entity) AND AN INTERPRETATION of the symbols of that statement (semantics) then the statement becomes true or false. — fishfry

Of course truth is relative to models. But mathematicians and philosophers most often take the liberty of saying "true" or "true in arithmetic" or "true statement of arithmetic" implicitly to mean "true in the standard model for PA". When a mathematician or philosopher says "it's a true statement of arithmetic", it is implicit that more formally that would be "is true in the standard model of PA".

Indeed, at least a few times in the past, I've mentioned to you that in greater formality, of course, sentences are only true or false per a given model or per a class of models (by the way, 'class' also is used informally (but such that with other locutions we can avoid 'class') there since, at least in Z set theories, there does not exist the proper class of all the models of a theory.

So when I say "true statement of arithmetic" you may always regard it at the usual shorthand for "true in the standard model of PA".

I should have counted on someone not recognizing that my remarks are merely in the spirit of personal reflection. — TonesInDeepFreeze

"When caught in a material error, I just claim I didn't really mean it that way."

Too tedious to likewise mock and debunk every single point you made in this post. I stand by my previous remarks. You made multiple substantive errors, which I corrected. You are the one who attacks every little technical inaccuracy anyone makes, even if their overall meaning is obvious. But you don't hold yourself to that same standard.

"When caught in a material error, I just claim I didn't really mean it that way." — fishfry

I said that I should not have said 'formally'. Recognizing an error in wording is not a bad thing. I don't blame you if you took my remarks not in the spirit I hope they would be received; but I have corrected a certain wording that I recognize now to be not what I meant. Actually, it occurs to me that an even better word is 'discursively' in the sense of 'analyzing'. If another poster did that in conversation with me, then I would welcome that as quite reasonable.

Again, my point was that I do see a difference in conceiving discursively and visualizing. And that, personally, I conceive of infinite sets discursively even though I can't wholly visualize them.

It's a sidenote in this discussion inspired by what I thought was an apt and well said comment by another poster. My take is not even remotely a technical or even philosophical thing.

Too tedious to likewise mock and debunk every single point you made in this post. I stand by my previous remarks. — fishfry

"When caught in a material error, I just claim that I was not in error".

/

(1) Rejecting the axiom of infinity does not entail that there are no infinite sets. Rather, both rejecting the axiom of infinity and adopting the negation of the axiom of infinity entails that there are no infinite sets. (This pertains to ZF.)

(2) It is not precluded that one may provide an axiomatization that proves that there are infinite sets other than by adopting the axiom of infinity.

(3) It is utterly ordinary that mathematicians and philosophers say "true statement of arithmetic" as implicit for "true in the standard model". If you don't believe me, then you could do an Internet search on 'true arithmetic' and see an article at your favorite go-to Internet encyclopedia. Moreover, you have seen many posts by me in which I was emphatic in explaining that, formally, truth is always per models.

(1) Rejecting the axiom of infinity does not entail that there are no infinite sets. Rather, both rejecting the axiom of infinity and adopting the negation of the axiom of infinity entails that there are no infinite sets. (This pertains to ZF.) — TonesInDeepFreeze

"During a lecture, the Oxford linguistic philosopher J. L. Austin made the claim that although a double negative in English implies a positive meaning, there is no language in which a double positive implies a negative. Morgenbesser responded in a dismissive tone, "Yeah, yeah.""

https://en.wikiquote.org/wiki/Sidney_Morgenbesser

I don't know the relevance you intend with that quote.

I don't know the relevance you intend with that quote. — TonesInDeepFreeze

"My point was to contrast informal deduction with "mind's eye" visualization."

You are the one who attacks every little technical inaccuracy anyone makes, even if their overall meaning is obvious. But you don't hold yourself to that same standard. — fishfry

(1) It's hyperbole that I "attack" in all cases. Rather, most often I just plainly state the correction. But often when it's a crank who continues to ignore the correction, then I do comment personally.

(2) It's hyperbole that I take exception to "every little" inaccuracy. I quite understand that in the confines of posting boxes, one can't always cross every technical 'i' and dot every technical 't' (spoonerism intended). I've remarked about that a few times. So I comment on lapses or liberties taken when (a) They are material, (b) They are slight but still the subject is markedly better represented by corrections. (c) I would like to suggest notation I think is sharper or significantly more elegant.

(3) I do hold myself to a standard. Indeed, when I post technical stuff, I almost always regret that my formulations and explanations still could be more rigorous. But, as mentioned, posting does not always allow such perfection. And when I have make a significant material error, I either correct myself in edit (stating that an edit has been made if the correction is consequential enough) or correct myself when brought to my attention. Indeed, in this instance, it was not even technical, but still material, that I erred by saying "formal" when what I meant is 'deductive' or 'discursive' even if not formal, and I posted that I recognized that error.

[Added in edit:]

even if their overall meaning is obvious — fishfry

Even if the gravamen is correct, there still can be points along the way that bear correction.

/

There is no simple formula for when corrections or suggestions for sharper formulations are due. So one has to "play it by ear" and use one's best judgment. I hope that I am corrected when appropriate but also hope that some reasonable liberty is granted, just as I hope that that is how I comment on others. And, as said, I virtually always kinda rue that it would be unwieldy for me to even make every formulation pinpoint given the context of posting. I understand that others might sometimes feel similarly about their own posts too.

Main thing is: If someone replies to me with a plain and correct correction, I do not (or at least I hope I do not) take that personally. And I think it's a lot better when others don't take it personally. Cranks, though, are a whole other ballgame.

but also hope that some reasonable liberty is granted — TonesInDeepFreeze

Was it not perfectly clear the other day that the usual order on the integers carries over to the extended integers? Where was your reasonable liberty then? Your concept of reasonable liberty only goes one way.

But today you were flat out wrong to claim that "reason" shows that there are infinite sets. That's an inaccuracy which no amount of reasonable liberty can fix. Likewise your invocation of unrestricted comprehension to justify that claim. I couldn't find any reasonable liberty there so I corrected your material errors. It's a full time job, let me tell you.

I don't get it. Or maybe adducing that quote is just your way of saying "yeah yeah" ironically. If so, whatever.

Was it not perfectly clear the other day that the usual order on the integers carries over to the extended integers — fishfry

Your question is answered by reading what I posted about that.

But today you were flat out wrong to claim that "reason" shows that there are infinite sets. — fishfry

You skipped what I said about that.

Likewise your invocation of unrestricted comprehension to justify that falsehood. — fishfry

Whoa. I did not at all invoke unrestricted comprehension. Not only is invoking unrestricted comprehension not in my words, not derivable from my words, but you would well know from many exchanges that you've seen that I have myself explained to other posters that unrestricted comprehension is contradictory. And I even explicitly wrote "with some restrictions".

I couldn't find any reasonable liberty there so I corrected your material errors. — fishfry

My error was to say 'formal'.

It's a full time job, let me tell you. — fishfry

You claim.

/

By the way, Dedekind himself gives an intuitive argument for the existence of infinite sets unrelated to axioms.

I don't get it. Or maybe adducing that quote is just your way of saying "yeah yeah" ironically. If so, whatever. — TonesInDeepFreeze

At this point I'm just trolling you. You're sometimes an easy target because you take yourself so seriously; and I was born a wiseass and can't help myself.

Whoa. I did not at all invoke unrestricted comprehension. — TonesInDeepFreeze

Heck you didn't. You, or somebody using your keyboard (do you have a cat perchance?) wrote:

I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle). — TonesInDeepFreeze

My emphasis in boldface.

That is unrestricted comprehension or my name's not Gottlob Frege. You wrote "with some restrictions," but the required restriction, namely restricted comprehension, would entirely negate your point.

At this point I'm just trolling you. — fishfry

A confession long overdue.

/

"I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle)." - TonesInDeepFreeze

The words "with some restrictions on that principle" explicitly declare that the idea is not unrestricted comprehension.

the required restriction would negate your point. — fishfry

The required restriction is that we don't claim that every property determines a set. That doesn't entail that there are not many properties that do entail a set. One of them may be the property of being a natural number. And (1) informally, in non-axiomatic reasoning, such as was my context, one may take the property of being a natural number to determine a set, without contradiction, and (2) Even formally, in ZF\AxInf, we have that AxInf is equivalent with "there exists the set of all and only the natural numbers". So even there, one may have the background of ZF\AxInf and also consider the property of being a natural number and think, "You know what, I think that property determines a set; in fact, I feel so strongly about it that I'm going to make it an axiom" and that in no way commits one to unrestricted comprehension.

If in one's mind, one views the property of being a natural number as suitable for determining a set, then one hasn't thereby committed to unrestricted comprehension. And, more generally, if one looks favorably on the notion that properties determine sets, albeit with some restrictions, one hasn't thereby committed to unrestricted comprehension. And this doesn't even have to be formal. I consider the property of being a book on my bookshelf and regard that property as determining a set.

I am not committing contradiction when I think "The property of being a natural number. Yeah, I think that is one of the properties to which the restrictions don't need to be applied, so, yeah, I do see that property as determining a set" while also being aware, "The property of being a member of oneself. Yeah, I've learned that that I better not allow that that property determines a set".


/

And Dedekind himself gave an argument for the existence of infinite sets without reference to axioms.

Without the axiom of infinity, THERE IS NO SET containing all the natural numbers. — fishfry

That is a good one to get back to.

Dropping the axiom of infinity does not entail that there is not a set that has all the natural numbers as members.

Rather, dropping the axiom of infinity from ZF and adding the negation of the axiom of infinity entails that there is not a set that has all the natural numbers as members. (To be exact, adding the negation of the axiom of infinity entails that there is not a set that has all the natural numbers, whether or not we also drop the axiom of infinity, but to not drop the axiom of infinity and add the negation of the axiom of infinity entails all sentences in the language anyway.)

Not a full time job, but a bit of labor it is.

"0 = the limit of the function 1/x where x ranges over the positive natural numbers," what do you mean by limit? Do you mean that for all epsilon there exists a delta such that ... etc? Where does your definitional parsimony end? If I say, "Let X be a topological space," should I replace that with, "Let X be a set along with a collection of its subsets satisfying ..." and then repeat the definition of a topology? — fishfry

Yes, I mean the ordinary definition.

Your analogy works in my favor. Given that I already have a definition of 'a topology', I can just say 'a topology' without reciting again its definiens. Given that I already have a definition of 'the limit', I can just say 'the limit' without reciting again its definiens.

'lim' is a variable binding operator, but it can be reduced to a regular operation symbol:

Df. If g is a function from the set of positive natural numbers into the set of real numbers, and there exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y, then Lg = the unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y

You see that 'L' is an operation symbol that takes only finitely many arguments - in this case, one argument.

The argument itself is an infinite set (an infinite sequence in this case), which is okay, because the operation symbol takes only finitely many arguments - in this case, one argument.

And we prove, regarding the function f we previously defined:

Lf = 0

In everyday parlance, "the limit of f is 0" [...] — TonesInDeepFreeze

So, we can boil it all down to a 1-place operation symbol, say 'L' and an argument, in this case 'f'.

Given that I already have a definition of 'the limit', I can just say 'the limit' without reciting again its definiens. — TonesInDeepFreeze

This is what I'm confused about with your objection to the extended reals (or integers, etc.)

I noted that the extended reals are essentially a notational convenience, and we could live without them.

And you seemed to be arguing that because we COULD live without them, then we SHOULD live without them.

That's the part I don't understand. The definition of limit is rather involved, at least for people first encountering it, involving epsilon and delta and universal and existential quantifiers and so forth.

By your logic (as I understand it), it would be parsimonious (a virtue, I gather, but I'm not sure why) to dispense with it, and do the raw epsilonics every time we want to mention a limit.

So that's what I'm genuinely confused about (no trolling, no jokes). You object to the use of the extended reals as a notational convenience for infinite limits and limits at infinity; but you do NOT likewise object to the use of the word "limit" as a notational shortcut for the epsilonic definition of a limit. Nor do you want me to further break that down by defining absolute values and the < sign.

I don't understand your point about the extended reals. They're a definition that makes expressing some mathematical ideas simpler, exactly like "limit" or "<" or "12" for "xxxxxxxxxxxx".

Why do you approve of some notational conveniences and not others? After all, ALL definitions are notational conveniences and could in principle be dispensed with in favor of writing out their meanings until we have nothing but primitive logical symbols. That would be maximally parsimonious yet also maximally incomprehensible. The whole point of definitions is to help people "chunk" their understanding of complex concepts. That's exactly what the extended reals do. They let us express infinite limits and limits at infinity in a compact notation.

This is what I'm confused about with your objection to the extended reals (or integers, etc.) — fishfry

I have no objection to the system of extended reals. I explicitly said that it is rigorous. And, of course, it can be useful rotationally (and perhaps, I don't know, it is required for certain aspects of mathematics). I'll add also that it is interesting

And I don't claim that one can't define 'extended integer'.

For my thoughts further about all that, I refer to the posts I wrote.

I noted that the extended reals are essentially a notational convenience, and we could live without them. — fishfry

I don't know enough to say that there aren't contexts in which we do need them as not mere convenience. But, yes, part of my point is that they are not required for the simple context in this thread.

And you seemed to be arguing that because we COULD live without them, then we SHOULD live without them. — fishfry

That is not my argument. For my actual points, I refer to the posts I wrote.

The definition of limit is rather involved, at least for people first encountering it, involving epsilon and delta and universal and existential quantifiers and so forth. — fishfry

It's not really very involved, but my guess is that it is involved enough (it has a universal quantifier followed by an existential quantifier followed by a universal quantifier) that its mild convolution scares students in the first week of calculus. Though, aha!, if one has learned basic symbolic logic, the quantifiers are a snap.

By your logic (as I understand it), it would be parsimonious (a virtue, I gather, but I'm not sure why) to dispense with it, and do the raw epsilonics every time we want to mention a limit. — fishfry

Indeed, you don't understand what I said. Indeed, you are saying the exact opposite of what I said. My previous post was that we do not have to reiterate the definition of 'limit' each time. Indeed, I showed that we can boil the definiens down to just a 1-place operation symbol and a single argument appended to the operation symbol.

I am genuinely baffled why so often in this forum you get me completely backwards.

You object to the use of the extended reals as a notational convenience for infinite limits and limits at infinity — fishfry

No, I don't. I pointed out problems with a particular formulation you used for sequences. And later in the back and forth between us, I mentioned that, more generally, the extended reals can usually be dispensed. I didn't say use of them is generally incorrect.

I don't understand your point about the extended reals. — fishfry

Perhaps you would go back to the posts and tell me the first sentence I said that you don't understand and that is not explained in subsequent passages.

parsimonious — fishfry

I thought that in context 'parsimonious' would be understood in reference to the point I made that the reals are sufficient for the entities for basic analysis, so it is less parsimonious to rely on yet more entities (-inf and +inf). But perhaps I should have been explicit about that. And again, that is not to say that the system of extended reals is incorrect.