You were wrong to claim I was not correct — TonesInDeepFreeze

No need to hammer something down when my previous post already agreed that, per set-theoretic context, you were correct. You can relax. I simply made the point that my misinterpretation was not unreasonable.

I reiterated the point that I was correct to support the additional point, which you did not mention, that I was also not arrogant about it.

I reiterated the point that I was correct to support the additional point, which you did not mention, that I was also not arrogant about it. — TonesInDeepFreeze

Sure. You were not arrogant about it. Extra brownie points.

Extra brownie points. — Kuro

Your defensive sarcasm is misplaced.

So I take it that restricting 'object' to refer only to abstractions is not acceptable to you. Thus, indeed you do not agree that abstractions are objects. Thus, indeed you contradict yourself when you also said: — TonesInDeepFreeze

Why do you conclude that? If I said, I'll only agree to call that type of thing which we sit on "a chair", if that's the only type of thing we will call "chair", would you conclude that it is not acceptable to me to call that type of thing a chair? What I said, is that we can call an abstraction an "object" if that's the only type of thing we call by that name. It's not uncommon to restrict definitions in this way, it makes deductive logic more productive by avoiding equivocation.

"I cannot agree that abstractions are objects" is tantamount to "abstractions are not objects".

"We restrict 'object' to refer only to abstractions" is tantamount to "only abstractions are objects".

So what you said is tantamount to: Abstractions are not objects unless only abstractions are objects. But you also deny that only abstractions are objects. Thus you affirm that abstractions are not objects. — TonesInDeepFreeze

This is a complete misunderstanding. Abstractions are a very specific type of thing. I have no objection to saying that abstractions are objects, if I reject other senses of "object" which I am familiar with, thereby naming a category "objects", and placing abstractions within this category. Just like in my example, we can name a category "chairs", and place the things we sit on in that category, and we can name a category "objects" and place abstractions there. However, since abstraction are such a unique, and very specific type of thing, I do not see how anything else could be placed in this category.

Or, we might realize that just like chairs are only one type of thing that we sit on, objects are just one type of abstraction. But if this is the case, then all objects are abstractions, but not all abstractions are objects. And we still don't have anything other than abstractions as objects.

I will though, on reconsideration allow the possibility that we could have a category "objects" and there might be things other than abstractions, which may be similar to abstractions in some way, which might also be placed in that category, "objects". So if you believe that there are other things similar to abstractions, which you believe ought to be placed in that category of "objects", you might demonstrate to me the reasons why you believe this.

I am happy you have finally found a number to use in place of infinity. You could show us how that works with the Lorentz factor. — jgill

But what is there left to move when the Cosmos arrives at its de Sitter heat death condition where all its degrees of freedom are embedded in its event horizon.

Time and change – as scaled by the Lorentz factor - have effectively ground to a halt. The only action is the quantum sizzle of the radiation attributed to the event horizon. We have reached the edge of the conformal disk in a finite fashion. The universe beyond may be supraluminally infinite. But it too is most likely to be in the same generalised condition.

So this may be the surprising thing. The reciprocality of the Planckscale start to the Comos means that a Big Bang with a minimum spatiotemporal scale and maximum energy density just simply turns itself inside out to become an equally finite Heat Death of maximum spatiotemporal scale and minimum energy density.

Lorentzian invariance would thus be emergent as particles formed with local momentum and inertial mass. And then it would dissolve from relevance as those particles get swept up and their energy radiated over the cosmic event horizon.

Of course, there is the little issue of the dark energy sustaining the whole show after the Heat Death. From that perspective, the Comos does expand forever and so "something" is always being added in terms of an infinite metric.

But once everything we physically could care about has reached its max ent state, do all those extra degrees of freedom actually count for anything? Even if there is potentially an infinite amount of them as dark energy repulsion just keeps stacking up in its supraluminal "biggest picture" way?

And it is still a remarkable thing if the cosmos had to find a way to close itself in this fashion so as to achieve concrete existence. It had to have a cut off at the beginning of the reciprocal kind that could once again be its cut off at the end.

:clap: Eloquently said!

But the problem with setting a largest number is that it rules out irrational numbers such as pi, sq-root 2 etc because they cannot continue to infinity as decimals and therefore become expressible as ratios. — unenlightened

Out of curiosity, do you have citations on this point? I would argue something somewhat different. But I'm interested if there are discussions that support your view here.

Agent Smith said that said set theory allows that a part can be equal to a whole. I correctly pointed out that that is not true. (For that matter, 'part and whole' are not even terms of set theory). And I correctly pointed out that what set theory does say is that in some cases a proper subset is equinumerous with its superset. — TonesInDeepFreeze

:up: Aren't odd numbers a part of natural numbers? Is it not true that the cardinality of the former equals that of the latter? :chin:

Eloquently said! — jgill

Stop it. Nothing makes me doubt what I just wrote more than someone's apparent agreement. :grin:

But I was thinking of you and rock climbing mathematicians the other day when I caught up with this sad story – https://profmattstrassler.com/2019/08/06/a-catastrophic-weekend-for-theoretical-high-energy-physics/

And also read this NYT story on how physicists (or at least mathematical physicists) are keen on mountaineering and bouldering - https://www.nytimes.com/2001/02/20/science/a-passion-for-physical-realms-minute-and-massive.html

I always thought from my experience that maths types were invariably into classical music. It was the physicists who scaled peaks.

For what its worth, I don't like a landscape that presents a technical challenge – one that has to be solved like a riddle. I like running fast and dangerously on twisty goat trails in the hills in a way that becomes quite unconscious. While listening to punk rock. A flow experience.

So I take it that restricting 'object' to refer only to abstractions is not acceptable to you. Thus, indeed you do not agree that abstractions are objects. Thus, indeed you contradict yourself when you also said: — TonesInDeepFreeze

Why do you conclude that? — Metaphysician Undercover

I take it that you don't take it that the only objects are abstractions, because you went on to say why you don't take it that the only objects are abstractions. You wrote:

"I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity." [Bold added]

The bold part is your argument why you don't restrict 'object' to refer only to abstractions.

Or, are you now saying that the only objects are abstractions?

Yes, I knew you would reply by shifting back around again from your own words but pretending that you haven't. This will go on indefinitely with you, as you play a silly game that is the forum equivalent to a child's peek a boo.

Aren't odd numbers a part of natural numbers? Is it not true that the cardinality of the former equals that of the latter? — Agent Smith

Yes. That has never been in question here. Indeed I reiterated just what you said in my post that you are replying to now! What is in question here is your your ignorant misunderstanding that equality of cardinalities in set theory implies that set theory says that a (proper) part can be equal to the whole. Please pay attention to exactly what you have said and what I have said.

Again, in set theory it is the case that infinite sets are such that they have proper subsets of the same cardinality as the set. But that is not at all to say that there are sets S and T such S is a proper subset of T and S is equal to T.

If T is infinite, then there exists a proper subset S of T such that there is a bijection between S and T (thus card(S) = card(T)). But there are no sets S and T such that S is proper subset of T and S = T.

By the way. In set theory we have these definitions:

T is infinite <-> T is not finite

T is Dedekind infinite <-> there is a proper subset S of T such that there is a bijection between S and T

In set theory, even without choice, we prove:

there is a proper subset S of T such that there is a bijection between S and T -> T is infinite

In set theory, with choice, we prove:

T is infinite -> there is a proper subset S of T such that there is a bijection between S and T

So, in set theory with choice, the definitions of 'infinite' and 'Dedekind infinite' are equivalent anyway:

T is infinite <-> T is Dedekind infinite.

Also, even without choice, we can prove that there do exist Dedekind infinite sets, such as your example of the odds and the naturals.

Most interesting. — Ms. Marple

Please continue posting but I suggest you engage with those who have the same level of understanding or higher as/than you. I'll read your posts as and when I can. Bonam fortunam TonesInDeepFreeze.

I will not impose upon myself a restriction from commenting on your posts.

I will not impose upon myself a restriction from commenting on your posts — TonesInDeepFreeze

No problemo! It only means I should improve the quality of me posts! Have an awesome day monsieur!

And the way for you to do that is to read a book on the subject.

And the way for you to do that is to read a book on the subject. — TonesInDeepFreeze

Thanks for the advice! On it!

On it! — Agent Smith

Oh really? What book?

Oh really? What book? — TonesInDeepFreeze

Recommend one, a simple one, to me. I have about 1.5 Terabytes of books, includes those on math.

I hadn't already recommended this?:

First:

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar

That is to get a solid understanding of the first order predicate calculus, which is the ground level for formal mathematics and formal philosophy.

Supplement:

Chapter 8 of Introduction To Logic - Suppes

That is for the best explanation of formal definitions I have found.

Supplement:

The introductory chapter of Introduction To Mathematical Logic - Church

That is for the very best overview of the subject of modern logic one would ever find.

Then:

Elements Of Set Theory - Enderton

Much obliged, señor!

De nada.

De nada — TonesInDeepFreeze

:up:

Sorry, no reference at all It just seemed obvious that if there is a largest number, decimal iterations must end at it - one cannot have the largest-number-plus-one-th decimal place. Thus no irrational numbers, Or to put it another way, the number of points on the number line between 0 and 1 will be finite.

But indeed so much would have to change that I cannot see how the numbers game would survive - most numbers would have no square for example, and calculations would keep ending in ERROR like on early calculators.

Well, let's look at this from the perspective of scale - I grok 2, I've encountered it; I also grok 38768, I've met this number (it's my bank account number, :joke: ); I, however, can't grok 10¹⁰⁰ (a googol), to do so I'd need someone to put that number in perspective i.e. it hasta be relatable in re my apperceptive mass, my experiences. In almost all videos on astronomy, scientists try to scale down the vastness of the solar sytem using tennis, soccer, basket balls, the size of football fields, etc. as points of reference we can easily grasp.

So, in my humble opinion, there's got to be a finite number such that it's, for all practical purposes, $\infty$ to us. For example, did I mention this already?, the speed of light (186000 miles/sec, a finite number) behaves like $\infty$ - we can never attain a speed of 186000 m/sec and that, in a sense, is infinityish. Wouldn't you agree?

and calculations would keep ending in ERROR like on early calculators. — unenlightened

Sounds like you can remember Texas calculators and Polish notation too. :up:

Choose the highest number you will allow

You would admit that someone else might choose a higher number.

And whenever someone chooses their highest number, someone else can choose one plus that number.

So which should we take to be the highest number we allow?

Mathematics should be limited to only the numbers you will personally allow?

/

Suppose there is a highest number any living human can now conceive in the manner you require. It is not ruled out that future humans may conceive higher numbers. So when that happens, we will reformulate mathematics to accommodate the new highest number? And if the person who conceives the highest number dies, then we'll reformulate mathematics to bring down the highest number? Then every day, we'll check the newspaper to see what the new highest number is?

/

Consider a music CD, with about 850 mb. Consider all the possible combinations of those 850 mb as zeros and ones. Big number. I can conceive it though, since I can conceive of changing every one of the bits on the disc. But I can also conceive of 8 billion discs - one for each person on the planet - and each of those dics having capability of changing every zero and one. A bigger number. But I can also conceive of each person on the planet having a 100 terrabyte computer to store lots and lots of 850 mb music albums, and then to switch the ones and zeros around on each of them as many ways as combinatorically possible. A bigger number. But I have to check with you each time I proceed to a bigger number whether you also can conceive it?

Don't you see that mathematics is general - not confined to only numbers that particular human beings can visualize in the way that you require at a particular time in the history of the universe.

For any natural number n, there is the natural number n+1. So there are infinitely many natural numbers. That is not thwarted by your limitations of what you can personally conceive at this particular moment.

we can never attain a speed of 186000 m/sec and that, in a sense, is infinityish. Wouldn't you agree? — Agent Smith

No, I don't agree.

The the number of states my lamp can be in is 2 - on or off. There is no counting past the number 2 when counting the number of states my lamp can be in. You can never attain more than 2 possible states for my lamp. So 2 is infinityish?

/

If 186000 is infinityish is then so is 186000 x 1000, since 186000x1000 m/millisecond is the speed of light. Etc.

So 2 is infinityish? — TonesInDeepFreeze

First time in my life I was able to count from one to infinity and then back. Thanks.

And it did not take forever, either.

Indeed, one can look at this from an individual's point of view, muchas gracias. I really wouldn't want to impose my own limitations on the world; that most assuredly would be one of the dumbest things a person could do.

However, the speed of light (186000 miles/sec) isn't personal; it's not something I wish and nor did I order it to be so. In fact I would like it not to be true, but that's a different story altogether.

My point is, from what I know of physics, no calculations on velocities at least will ever exceed 186000 miles/sec. In other words if there's a claculator dedicated to calculation velocities, any result for speed that exceeds 1860000 miles/sec should return ERROR.