Infinite divisibility doesn't entail a contradiction. Rather, infinitely divisibility along with the other premises entails a contradiction. Moreover, you are adding another premise (call it 'DT'): if infinite divisibility, then tasks can be performed at each of the infinitely many times. Therefore, we are entitled to question any of the premises, including the new one DT, not just infinite divisibility.

An interesting point is that while we can express the indiscernibility of identicals as a first order schema, we can express the identity of indiscernibiles as a first order schema if and only if there are only finitely many operation and predicate symbols.

It's an interesting exercise to try to express the identity of indiscernibiles as a first order schema with a language of infinitely many non-logical symbols. You'd think you'd just reverse the indiscernibility of identicals. But when you try, it doesn't work! If I'm not mistaken, one of the famous logicians proved it can't be done.

/

Another nice thing: Identity theory can be axiomatized another way, courtesy of Wang:

For all formulas P:

Ax(P(x) <-> Ey(x=y & P(y)))

From that we can derive both the law of identity and the indiscernibility of identicals.

If you are doing first order logic, how do you quantify over all propositions P? — fishfry

We quantify over them in the meta-theory not in the object theory.

That is what an axiom schema is.

For example (leaving out some technical details here:):

In first order PA the induction axiom schema:

For all formulas P:

(P(0) & An(P(n) -> P(Sn))) -> An P(n)

In set theory, the axiom schema of separation:

For all formulas P:

AzExAy(y e x <-> (y e z & P(y)))


Those are statements in the meta-theory that describe an infinite set whose members are all axioms that are in the object-theory.

"Identity theory" is first order predicate logic with equality. Is that your own terminology? — fishfry

I stated explicitly several times that that is what I mean by 'identity theory'. I recall having seen the term used professionally before, and so I adopted it a long time ago, but I would have to dig to find citations. I like it, because it is a first order theory about one certain predicate that is indeed the identity predicate.

If someone says "I'm talking about blahblah theory'" and they tell me the axioms, then I don't quarrel with them about it. I know the axioms so I know precisely what is meant by 'blahblah theory'.

Benacerraf:

"A. Aladdin starts at to and performs the super-task in question just as
Thomson does. Let t1 be the first inistant after he has completed the whole
infinite sequence of jabs - the instant about which Thomson asks "Is the
lamp on or off? - and let the lamp be on at t1.

B. Bernard starts at to and performs the super-task in question (on an-
other lamp) just as Aladdin does, and let Bernard's lamp be off at t1.

I submit that neither description is self-contradictory, or, more
cautiously, that Thomson's argument shows neither description to
be self-contradictory"

But that contradicts:

If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.

and

If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.

Benacerraf is saying the lamp gets switched in a way that is not possible given Thomson's conditions.

When we describe the events, we have to look closely and exactly at whether they may occur given Thomson's premises. We can't dissolve Thomson's argument merely by ignoring the premises of the argument.

/

Benacerraf:

"According to Thonmson, Aladdin's lamp cannot be on at t,
because Aladdin turned it off after each time he turned it on.
But this is true only of instants before tl!"

There he does recognize the premises, but, it seems to me, he mistakes them. The premises don't cover just what happens before 12:00. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.

"Nothing whatever has been said about the lamp at t1
or later."

That seems to me to be incorrect. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.

Benacerraf:

"The explanation
is quite simply that Thomson's instructions do not cover the state
of the lamp at t1, although they do tell us what will be its state at
every instant between to and t1"

The instructions don't need to specify what happens at 12:00. The instructions specify what happens at all moments and also what happens before 12:00, but what happens at 12:00 still must conform to the instructions that apply to all moments.

The issue is not that the instructions don't specify what happens at 12:00. The issue is that the instructions entail that at 12:00 the lamp is Off and at 12:00 the lamp is On. Thus the instructions are contradictory.

If Benacerraf is not skipping the condition, then where does he recognize it? [EDIT: Actually he does address it, but, as far as I can tell, he gets it wrong when he addresses it.]

What essential difference is there between Aladdin/Bernard and Cinderella?

The converse of extensionality is not provided by the law of identity. It is provided by the indiscernibility of identicals.
— TonesInDeepFreeze

you are making a point I can't agree with.

I take your point about set equality as expressed in the Wiki page on extensionality, which says, "The axiom given above assumes that equality is a primitive symbol in predicate logic."

If you mean something else, we're back to square one. — fishfry

'=' is primitive.

But there is more to say.

So indeed, let's go back to square one:

'=' is primitive in logic (first order logic with equality, aka 'identity theory').

And '=' has a fixed interpretation (which is semantical, not part of the axioms) that '=' stands for identity.

So identity theory has axioms so that we can make inferences with '='.

The axioms are:

Ax x=x ... the law of identity

And the axiom schema (I'm leaving out technical details):

For all formulas P:

Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals

Then set theory adds its axiom:

Axy(Az(zex <-> zey) -> x=y) ... extensionality

Now we ask how we derive:

(zex & x=y) > zey

Answer: from the indiscernibility of identicals. Indeed the above is an instance of the indiscernibility of identicals, where P(x) is zex.

I think those examples and a common informal context are okay. They suggest that, for example, a rock is not a set. My point is that it would only be a far stretch of the notion of 'set' that would permit taking a rock to be a set.

@Michael

Next would be to examine whether your inference is correct that the problem shows that time is not infinitely divisible (or that it is not possible that time is infinitely divisible - and the modality there may make this more complicated). If I understand correctly, Thomson does't announce such a view about time, though, of course, what Thomson may believe doesn't determine our own conclusions.

@Michael @fishfry

I haven't yet read all of Benacerraf's paper, but at least where he disscusses Aladdin and Bernard, it seems to me that he's not addressing Thomson's problem but only offering a different problem that does have an easy solution.

With Thomson's problem we have:

If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.

and

If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.

It seems to me that Benacerraf is skipping that condition. And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf.

The ‘objects’ or ‘elements’ that constitute a house: walls, ceiling, windows, door, etc. — javi2541997

The same set can be specified in two different ways:

{the door, the floor, the roof ... the balcony} [fill in '...' with all the other features of the house .]

{x | x is a feature of the house}

Without these elements or 'objects', the principal thing (the house) is senseless — javi2541997

Okay.

these three elements are necessarily elements of the house. — javi2541997

In casual conversation, the word 'elements' can be used that way. But if we are talking in a focused context about sets, 'elements' refers to members of a set. And the house is not a set. Sure, in some informal way, we could stretch the meaning of 'set' so that in some view a house is a set. But in a focused sense of 'set', a house is not set, just as a rock is not a set.

If anything at all were a set - a house, zebra, rock, cloud - then 'set' wouldn't have any special meaning. You could point to a stop sign and say, "Hey look at that set over there, the stop sign". But that's not the common notion of 'set'.

@Michael

Two presentations that are equivalent.

I would like to know how C2 and C3 are derived in Michael's version. That is rC1 and rC2 in my version.

But we get them anyway from my premise rP6 (the antecedent of Michael's C6). I'll show that presentation too (PRESENTATION2). And I think it is closer to Thomson's argument.


MICHAEL'S PRESENTATION

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00

From these we can then deduce:

C1. The lamp is either on or off at all tn >= 10:00
C2. The lamp is on at some tn > 10:00 iff the button was pushed at some ti > 10:00 and <= tn to turn it on and not then pushed at some tj > ti and <= tn to turn it off
C3. If the lamp is on at some tn > 10:00 then the lamp is off at some tm > tn iff the button was pushed at some ti > tn and <= tm to turn it off and not then pushed at some tj > ti and <= tm to turn it on

From these we can then deduce:

C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction]


TONESINDEEPFREEZE'S PRESENTATION:

Premises:

rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.

Conclusions:

rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.

Premise:

rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.

Conclusion:

rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.


Again, I don't know how we derive Michael's C2 and C3 (my rC1 and rC2). But we don't need them anyway:


TONESINDEEPFREEZE'S PRESENTATION 2:

Premises:

rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.

Conclusions:

rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
rC1: The lamp is neither Off nor On at 12:00. Contradicts rP1.


So, we don't have to be concerned whether rP1-rP5 entail rC1-rC3. Rather, we see easily that rP1-rP6 entail rC1-rC3. It's a clean and correct inference that way.

So, unless we do have a proof of Michael's C2 and C3 from his P1-P4, he has his argument out of order: we need my rP6 in the premises. And that seems to be flow of Thomson's argument too.

Relax! — Ludwig V

I was quite relaxed when I provided the information.

I was struck by the point that "at all times the lamp is either Off or On" appears to be true while "the lamp is neither Off nor On" appears to be false, by reason of a failed referent. It's true by definition that a lamp is either off or on, so if some object is capable of being neither off nor on is not a lamp. The story is incoherent from the start. We cannot even imagine it. — Ludwig V

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. You can go back to the argument to witness it step by step. Best is to read Thomson's paper that is not long and not abstruse, free to download.

Can you refer me to a source? — Ludwig V

I won't refer you to a source.

I'll refer you to this:


Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).

Let f(k) = SUM(j = 1 to k) 9/(10^j).

Show that lim(k = 1 to inf) f(k) = 1.

That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e.

First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k).

Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k).

Inductive hypothesis: 1 - f(k) = 1/(10^k).

Show that 1 - f(k+1) = 1/(10^(k+1)).

1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)).

By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)).

Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)).

So by induction, for all k, 1 - f(k) = 1/(10^k).

Let e > 0. Then there exists n such that, 1/(10^n) < e.

For all k > n, 1/(10^k) < 1/(10^n).

So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n) < e.


(I saw an argument in a video that is much simpler, but I didn't get around to fully checking out whether it's rigorous. But arguments that subtract infinite rows are handwaving since subtraction with infinite rows is not defined.)

If that isn't a house — javi2541997

I didn't say that a house is not a house. I said a house is not a set.

But what is the sense of doing those things separately? — javi2541997

What things separately?

I can't envision a house without a wall or a ceiling as structural elements. — javi2541997

Nor can I. That doesn't entail that a house is a set. Again, a house is a thing you live in. You don't live in a set; you live in a house.

Perhaps I am misunderstanding the concepts of "set," "order," "members," — javi2541997

Sets:

S = {the door of the house, the roof of the house, the floor of the house ... the balcony of the house}.

S is not a house. It is a set whose members are features of the house.

Members:

The members of S are the features of the house.

Order:

<door, roof, floor ... balcony> is one order

<floor, balcony, door ... roof> is another order

I've discovered that potential infinity is the definition of the sequence and actual infinity is the completion of the sequence. — Ludwig V

The adjective 'is potentially infinite' has no mathematical definition that I know of, including in alternative theories.

The adjective 'is infinite' is defined in mathematics.

The adjective 'is actually infinite' has no mathematical definition that I know of, including in alternative theories, unless it means simply 'is infinite'.

'is potentially infinite' is a notion about mathematics.

'is actually infinite', if not meaning simply 'is infinite', is a notion about mathematics.

I think it is just a result of thinking that you can write probability = 1 — Ludwig V

Who says anything about probability when merely mentioning that .9... = 1.

We prove that .9... = 1, from the definition of the notation '.9...'.

'.9...' stands for the limit of a certain sequence, and that limit is 1.

Anyone is free to regard '.9...' with a different definition and to get different results accordingly. But in context of the ordinary mathematical definition, we prove that .9... = 1.

If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. — Ludwig V

By the premises, there is no third state. Indeed, even if not a premise but a definition:

Df. 'On' means 'not Off'

there is no third state.

Or do you mean that it is not defined as on or off — Ludwig V

No, Thomson's argument is: The premises entail that at 12:00 the lamp is neither Off nor On, but the premises also include the stipulation that at all times the lamp is either Off or On, so the premises are inconsistent.

which leaves the possibility that it must be in one state or the other, we just don't know which. — Ludwig V

No, it's not a matter of knowledge. Rather, at 12:00 the lamp is neither Off nor On, which contradicts that at all times the lamp is either Off or On.

Not everything that matters is calculable. — Joshs

I should have said that that I don't know what his comment to me is supposed to mean in relation to anything I've written.

The crank writes, "[TonesInDeepFreeze claims] that the elements of a set may be concrete objects."

If the elements cannot be concretes and can't be abstractions, then what can they be?

Or does the crank reject even the notion of sets and elements?

Is a rock not a concrete? If a rock is not a concrete, then what is an example of a concrete?

If there is no such set that is the set of rocks on my table, then what are examples of sets?

/

The crank writes, "The sense of humour leaves the head sophist [TonesInDeepFreeze] exposed, revealing no control over the inclination to equivocate."

I dare not ask what in the world that is supposed to mean.

By the way, though I am a magnitude of light years away from being a sophist, I would rather be merely a sophist than a crank, since being a crank includes being a sophist and a lot worse too.

Set consisting of three balls colored red, white and blue. They also have differing weights. What is THE order? Just curious.
— jgill

The order is how items are organised with one another based on a specific attribute. The only distinguishing feature is that they are spherical. The weight and colours are only accessories. The set would be spheres, and the order would be the three balls. Right? — javi2541997

If the attribute is color then there are six orderings based on that attribute:

red ball, white ball, blue ball
red ball, blue ball, white ball
blue ball, red ball, white ball
blue ball, white ball, red ball
white ball, red ball, blue ball
white ball, blue ball, red ball

There is not just one ordering that we can call "THE" ordering.

If the attribute is size, and it is not the case that there are two balls with the same size, then there are six orderings based on that attribute:

the largest ball, the middle sized ball, the smallest ball
the largest ball, the smallest ball, the middle sized ball
the middle sized ball, the largest ball, the smallest ball
the middle sized ball, the smallest ball, the largest ball
the smallest ball, the largest ball, the middle sized ball
the smallest ball, the middle sized ball, the largest ball

There is not just one ordering that we can call "THE" ordering.

Do you see that?


By the way, this pertains to linear orderings (aka 'total orderings'). There are other kinds of orderings, especially partial orderings, but here the context is linear orderings.

ground, bricks, walls, ceiling, windows, and a door altogether make a set, which is the house. — javi2541997

I would think of those as aspects of the house, not members of the house. I wouldn't think of a house as being a set. There are sets of aspects of a house. But that set is not a house. A house is something you live in and pay a mortgage on. You don't live in a set and pay mortgage on a set. There is a housing market and a housing shortage; there is not a sets market and a sets shortage.

Questions are not interruptions. And no level is required to ask questions.

When someone lies about your posts and incessantly posts disinformation about the subject, then it is appropriate to comment on that and it is pertinent too to comment on the modus operandi behind it. After correcting a crank over and over and over, with the crank still continuing to post the disinformation and indeed adding even more, the more salient subject becomes not the topic but the deleterious effects of the crank. The point is to not normalize cranks. I see forums ruined by being inundated by cranks, not ruined by informed people posting back against cranks. Then, also, search engines include posts with disinformation from cranks near the top of search results on various subjects. And that is added to the disinformation and confused presentations about mathematics on Wikipedia. And that's along with the outrageous disinformation and confusion transmitted at the speed of light by AI answer bots. Thus the torrent of effluvia from which the Internet spirals down and down into a cesspool of disinformation.

I came to this topic hoping to learn anything about set, order, infinite — javi2541997

There's plenty of detailed information and explanation posted in this thread.

If you have any questions, or wish to learn more, then it's as simple as asking (and not asking someone who doesn't know anything about the subject).

Your positions and answers are quite good — javi2541997

Which positions? You think it's a good position to deny that a set with more than one member has more than one ordering?

The incessant crank says, "Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers."

There is no sophistry on my part. And no "replacing". I merely pointed out that proving that time is not continuous does not prove that time is not densely ordered (or infinitely divisible).

And the crank is so ignorant and mixed up about this very thread that he wildly infers that my remarks about the thought experiment vis-a-vis Michael's version of it entail that I have myself made certain claims about time beyond that not-continuous does not imply not-dense.

The garbage posting crank doesn't know what he's talking about, regarding continuity or denseness, or me. He is a bane upon reasoned inquiry.

/

The crank is on about .9... Whatever he's trying to say, in his usual thought salad way, we should at least recognize that the notation '.9...' is informal for the limit of a certain sequence.

Meanwhile he has his own utterly mixed up notions about what 'identical' and 'equal' mean. But he hasn't the least reasonability even to understand that his own having notions about what words should mean doesn't entail that everyone else is wrong for using the words both in their ordinary English senses and also in stipulated mathematical senses. He does not understand even the notion of stipulative definition, just as, in another instance, he does not understand even the difference between use and mention.

The crank falsely rails on and on against mathematics and modern logic, even though he has not read page one in a textbook in the subject. As he serves as a textbook example in crank sophistry.

The crank's latest posts are again a welter of blatant sophistry. If only one's time were infinite to write out out all that should be said about his confusions, illogic, self-contradictions, and lies.

@fishfry

The converse of extensionality is not provided by the law of identity. It is provided by the indiscernibility of identicals.

The crank says, "The deep stuff gets booted off the main page, being for most, undistinguishable from shit."

The crank can't discern irony, even when it is declared with an emoji.

/

The crank says, "tim is even lower down than you are. And both of you make a snake appear like an angel."

That's lower than lame.

Frame that.

'Not everything that counts can be counted, and not everything that can be counted counts' — Wayfarer

I haven't the foggiest what that is supposed to mean.

Here is a quote from Reddit that brings some clarity to the subject of "truth" in mathematics these days:

"Godel's completeness theorem, applied to group theory, says that any statement that's true for every group can be proved from the axioms of group theory. Similarly, there is more than one model of ZFC. The existence of various models of ZFC is analogous to the existence of different groups. Some statements are true in one model of ZFC and false in another. Such a statement is independent of ZFC." — jgill

(1) The completeness theorem is: If a sentence is entailed by a set of premises then the sentence is provable from that set of sentences. Or, equivalently, if a set of sentences is consistent then it has a model.

But 'a theory is complete' means that every sentence in the language for the theory is either provable in the theory or its negation is provable in the theory.

Now, I'm not sure, but I doubt that (first order) group theory is complete.

What does "true for every group" mean? Sentences are true or false in models. So does"true for every group" mean "true in every model of first order group theory"?

If yes, then, yes every sentence that is true in every model of group theory is provable in group theory, as follows:

If a sentence S is not provable from a consistent set of axioms G, then G plus ~S is consistent, as follows: By the completeness theorem, it is not the case that every model of G is a model of S. So there is a model of G that is also a model of ~S. So G plus ~S is consistent. Now suppose a sentence S is true in every model of group theory. But suppose it is not provable in group theory. So ~S is consistent with the axioms of group theory. So, by the completeness theorem, the axioms of group theory plus ~S has a model. But since S is true in every model of group theory, ~S is false in every model of group theory, which contradicts that there is a model of the axioms of group theory plus ~S.

(2) Yes, if ZFC has a model M then there are other models of ZFC that are not isomorphic with M. And, yes, there are sentences independent from ZFC. But I don't know what exact claim is made with "The existence of various models of ZFC is analogous to the existence of different groups." There's nothing notable about the fact that there are different groups. Since we have Lowenheim-Skolem, it's not even notable that there are non-isomorphic models of group theory.

proof implies truth, but truth does not imply proof.

Suppose we have a consistent set of axioms for mathematics (the set theory axioms will do nicely). Then if the axioms are true then all theorems derived from those axioms are true. But there are truths not derivable from the axioms.

In other words, whatever is provable is true. But it's not the case that whatever is true is provable.

@Michael: I see now that a few of my previous comments, while not incorrect, were not helpful for understanding the problem. I'm still at provisional stages, but this is what I'm thinking now:

I would organize Thomson's argument differently from the way he organizes it. Near the end of his argument, he says "But the lamp must be either on or off." But he's actually invoking a premise. It is natural to regard the lamp as being either Off or On and not both, but in this highly hypothetical context, it would be good to say that as an explicit premise.

Then Thompson invokes infinitely divisible time. But not as a premise. I would include it as a premise. The advantage of doing that is that then the premise is explicitly a candidate for rejection to avoid the contradiction.

I simplified the language of your conclusions (we don't need all those tn, ti, tj and inequality symbols), and I don't think you need the conclusions to be biconditionals to derive that the lamp is neither Off nor On.

('r' for 'revised')

Premises:

rP1: At all times, the lamp is either Off or On and not both.

rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.*

*The pushing of the button and the change are together instantaneous, and the button can be pushed only once in any moment. This is not needed except to simplify the argument (especially to state rC1, rC2 and rP6).

rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.

rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.

rP5: The lamp is Off at 10:00.

Conclusions:

rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.*

*Notice that T1 and T2 are in chronological order.

rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.*

*Notice that T1, T2 and T3 are in chronological order.

Premise:

rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.*

* We could easily make that mathematically rigorous.

Conclusion:

rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.


QUESTION: How do you state the arguments for rC1 and rC2 from the premises rP1-rP5?

Ugh, the crank drags me into his personal dispute by dissing me with passive aggressive faint praise as a way to diss the other poster. What a snake.

@fishfry: Probably some of these points you already know ; I'm mentioning them just to fill out the picture.

In Peano arithmetic (PA), we generate all the natural numbers with two rules:

* 0 is a number; and

* If n is a number, then Sn is a number, where S is the successor function. — fishfry

If PA here is first order, then PA does not have a predicate 'is a number' nor those axioms.

use the successor function to define "+" — fishfry

Just to be clear, that occurs in set theory, not in PA. In PA, '+' is not defined. It is primitive.

There is no "completion" of the sequence thereby generated, 0, 1, 2, 3, 4, ...In particular, there is no container or set that holds all of them at once. — fishfry

Of course, that's correct regarding PA.

We can do a fair amount of number theory in PA. We can NOT do calculus, define the real numbers, define limits, and so forth. — fishfry

Right.

In PA we have each of the numbers 0, 1, 2, 3, ... but we do not have a set of them. In fact we don't even have the notion of set. — fishfry

Right.

The axiom of infinity actually defines what we mean by a successor function for sets — fishfry

The axiom of infinity does not define anything, including the successor operation.

The successor operation only requires pairing and union:

Df. the successor of x = xu{x}.

That is logically prior to the axiom of infinity. Then the axiom of infinity only says that there is a set that has 0 and is closed under successor.

Then we prove that there is a unique set that is a subset of all sets that have 0 and are closed under successor.

Then we define w = the set that is a subset of all sets that have 0 and are closed under successor.

and says that there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X. — fishfry

Not "and". All it says is what you said after the "and": "there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X".

lets us construct a model of PA within ZF; and we take that model to be the natural numbers. — fishfry

Rather than "the model" I would say "the standard model". There are other models too. And models not isomorphic with the standard model.

PA gives you each of 0, 1, 2, 3, ... — fishfry

the axiom of infinity gives you {0, 1, 2, 3, ...} — fishfry

Both are right, and well said. In both PA and Z without infinity (even in Z with the axiom of infinity replaced by the negation of the axiom of infinity), we can define each number natural number, and in Z we can prove the existence of the set of all and only the natural numbers.

Show me your balls and I will tell you their order. — the crank

Yikes.

It was merely a math quip.

The crank enacts one of the starkest examples of mentally pathological illogic I've seen in a while:

I say, in clear, emphatic, and unequivocal terms that the 24 orderings are different orderings. It is at the heart of my point that they are different orderings. It would be absurd to say that they are not different orderings. But the crank says that I say that they are the same ordering.

Illogic doesn't get much more dire than the crank's.

It is not any more a contradiction for a set to have more than one ordering than it is a contradiction for a person to own more than one hat.

It's not a matter of whether I explicate the difference between concrete and abstract. Rather, whatever one's explication of the difference, abstract objects can be elements and concrete objects can be elements.

The crank argues by persistently ignoring the rebuttals, examples and explanations given him:

The crank says, "If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order."

That was answered posts ago by me. The crank can't or won't read the posts he replies to.

The crank asks, "Why did you say I lied about this?"

Here are the lies:

"You mix up physical objects and mathematical objects as if there is no difference between them"

I explicitly mentioned that a number is abstract and a rock is concrete. And I reiterated that. The crank is a liar.

"For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two." [the lie bolded]

I have never conflated abstractions with concrete. The crank is a foolish liar.

"[Tones In Deep Freeze] has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same"

I do not at all make any such utterly ridiculous statement that the orderings are the same. And I reiterated that. The crank is a profoundly illogical liar.

That needs work.

It leaves out that for the most used overall system for mathematics, it is not the case that every truth is provable.

It leaves out that the concept of mathematical truth is actually not formulated in terms of proof. Rather, proof and truth are formulated separately, but then mathematics shows that, for first order logic: A statement is provable from a set of premises if and only if the truth of the premises entails the truth of the statement.

It leaves out that the greatest objectivity is in the fact that it is machine checkable whether, at least in principle, a given formal sequence that is purported to be a proof is actually a formal proof.