Comparing infinities using different measures (density vs cardinality) that contradict each other proves the mathematicians don't have any idea of what infinity is — Gregory

Who is comparing? Please show how the two compare. How do they contradict each other?

https://www.encyclopediaofmath.org/index.php/Density_of_a_set#Density_of_a_measure

Wiki: In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X.

Wiki: The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A.

What is TS? — fdrake

Topological space.

On every page there is a description of a single particle, where it is, what is doing at the given time. Collectively all that information describes everything that exists and will ever exist. — Zelebg

Let's say particle alpha is under consideration. We measure time in seconds. Page 1, present time. Page 2 , 1 second from now. Page three, 2 seconds from now, etc. Page N, N-1 seconds from now. You would have to assume time stops at some point in the future in order to secure your "proof." So you would postulate that time is finite. But this seems to be part of what you wish to prove.

I must be missing some important debate points here, in my old age. :gasp:

Therefore, the total number of unique bits of information is finite, or there is some kind of information your monitor can not display, for some reason. — Zelebg

Say each piece of information is a string of alphabet symbols. Since the length of these strings is unbounded, so is the amount of information. We're not talking about computer programs that terminate.

Do you know what kind of properties a space would need to have so that every subset of it could be covered by a finite set of polygons? — fdrake

Compactness? Are your "polygons" abstract entities? Topological spaces or what? Compactness in TS if you adjoin limit points, I suppose. :nerd:

You don't get it, you can zoom in as much as you wish in arbitrary small steps. — Zelebg

Wow! So there are an infinite number of pixels in each photo. You're correct. Guess I don't get it.

You can also forget photographs and imagine all the knowledge there is about everything that will ever be is simply written in English words, with illustrations and diagrams. — Zelebg

There are a finite number of existing English words, but if there is no limit on the length of a word you could be speaking of infinities. There are 26 letters, so how many "words" could be constructed, of any length? Of two letters: 676. It goes up from there. 26^n. n increases without bound. Also, there could be an infinite number of diagrams.

Is the map the territory?

Well, no, the number of distinct digital photos of a given resolution is finite. But so what? — SophistiCat

Of course. The entire process is faulty. The assumption that every aspect of the universe can be so pixeled assumes his conclusion.

It's interesting to note that the duration of time has seemingly decreased throughout history. — 3017amen

Please cite a legitimate reference. Is this claim related to the observation that time seems to pass more rapidly as we grow older? A psychological phenomenon.

To terminate this discussion let me reveal that on my desk is a small Egyptian box, 5cmX3cmX1cm, in which I keep both zero and infinity. So you have my word that, yes, they do exist. :cool:

Yet the number of all those possible photos is not infinite. — Zelebg

Faulty hypothesis. :yawn:

It's a bridge though to Eliatic realms however, a secret door though that mathematicians don't know about — Gregory

Wiki: "The Eleatics maintained that the true explanation of things lies in the conception of a universal unity of being. According to their doctrine, the senses cannot cognize this unity, because their reports are inconsistent; it is by thought alone that we can pass beyond the false appearances of sense and arrive at the knowledge of being, at the fundamental truth that the "All is One""

More like a trap door IMHO. :roll:

Yes, it's difficult to square what appears as a very dynamic universe with block structure. All the mathematics I dabble in involves patterns of movement, so I am biased. :cool:

At this point all our intuitions fail, and we must adopt some form of relativism or say we know nothing of math whatsoever. — Gregory

It's not that bad. Really. :roll:

Here's where you need someone who has experienced this kind of enlightenment. I engaged in Zen practice for a short time a lifetime ago, but did not reach that stage. A friend who has had the experience says one's "I" perspective drops away and there appears to be an empty stage of awareness, a "no-thingness", and the various koans are comprehended (?). "Form is emptiness and emptiness is form" - that sort of thing.

This is an amateurs' forum, so one doesn't expect credentialed experts to spend time here. Still, with over 6K members there might be one who has real Zen experience?

For some reason this thread is not coming up on the forum page for me.

How about it, Dan? Get your degree? :smile:

As Tristan L has so generously and perspicaciously explained, the proof adduces a one-to-one function from the naturals onto the odds, and that is all that is needed. — GrandMinnow

This should put an end to the issue. But it won't. :roll:

I suggest someone who is a proponent of "more leisure time" try starting a GoFundMe account. This might provide a "peoples'" perspective on the idea.

My two areas of relative expertise are mathematics (classical complex analysis) and climbing. Regarding the latter, I have seen the American Alpine Club shift its priorities over the years, from preserving history and sponsoring Himalayan expeditions to raising money to give to youngsters so they can go play on the rock. And some of those young climbers avoid medical insurance, assuming a GoFundMe account will pay for their injuries. More supported leisure time. I compare this with the environment sixty years ago, when a friend who was living day to day, a "dirtbag" climber, rose on his own merits over the years to become a California billionaire.

However, even in a good economy like the present with plenty of jobs, many young people live in the "gig" world, moving from one temp job to another, with limited benefits. I see this in the academic world where tenure-track positions become adjunct appointments, with virtually no benefits.

So, I have mixed feelings on the issue. :chin:

How devastated and crushed and angry and heartbroken they're all going to be when the DNC steals the nomination from Bernie — fishfry

And how they will not vote next November, to show their frustration.

I'd suggest Zen Buddhism, but it's more a practice than a philosophy.

Everyone is dying to live in Alabama, .... — Xtrix

https://matadornetwork.com/life/10-reasons-moving-alabama-right-now/ :razz:

The cost of housing may be going up in many areas that have been reasonable in the past. I live in a nice middle class prairie community south of Colorado Springs with a population of about 40K, ten miles from a city of 100K where half the population is on medicaid. Until a couple of years ago one could buy a fairly new three bedroom house with three bathrooms on 2.5 acres with unrestricted views of Pikes Peak and the Wet Mountains for $250K. Yesterday I saw a house down my street go up for sale for $450K. This house would probably go for a million bucks in a comparable region of California.

Colorado has become a blue state, and our governor admires and wants to emulate California. May the saints preserve us . . .

As far as I am concerned, the only thing that really helps, are good examples, which are almost always lacking — alcontali

Good point. The only trait I share with R. Feynman is the process of learning a difficult concept by looking at examples. There are many mathematicians who can understand abstract notions very quickly simply by the words and symbols describing them. I envy them this ability. By studying examples it all clicks into place for me.

Many years ago I knew of a graduate student who spent a couple of years working up his PhD research project on a class of functions. Then one day he was asked for an example. A fruitless endeavor, for he soon learned the set of such functions was the empty set. :worry:

The conversation has drifted into ergodic theory rather than entropy. :roll:

That article does go on and on. I burned out after a short spell. Fifty years ago I was formulating my dissertation and was stuck at a difficult point involving a kind of infinite iteration. As I was lying in bed one night, beginning to drift off, I suddenly envisioned a geometrical image that provided a key to completing both the structure and proof of a difficult theorem. I still do a bit of research and imagery is important, although it's rare that a proof itself relies on geometrical figures.

At one time it was required that a student take a course in analytic geometry prior to an introduction to calculus. All the imagery in that course provided a bridge of understanding for calculus. Nowadays, that material is squeezed into calculus, and there are even calculus curricula that attempt to minimize imagery by going into n-dimensional vector spaces immediately.

If there are an infinite number of universes, there is one where the contents of this entire forum were created by monkeys randomly typing at keyboards. — Douglas Alan

A countable or uncountable number of universes? Would there be a difference? :chin:

Assuming equal probability, the finite number of states simply means that the fewer the states the higher probability one re-occurs over a lengthy series of experiments. And as the number of states increases without bound ("goes to infinity"), the probability of a particular state shrinks toward zero. Roughly speaking.

The function has to do with the mixing of the particles, say, from moment to moment. Without preservation of "area" the distance between two points might shrink each iteration, and the configuration one would like to see re-emerge would not be possible. Roughly speaking.

By "intelligent design" one seems to imply some sort of entity that "causes" that phenomenon. But what if the design appears to be a product of intelligence when in fact it may not be? Is there effect without cause? All most of us ever see in this world is effect resulting from a cause, or a host of causes. Thus, we question, What was "before" the Big Bang? Suppose the Universe is different, but we "see" what we are accustomed to seeing. Then time reversal in mathematical equations in physics might describe reality better than normal perspectives. We simply can't see or process what is "really" out there.

Just babbling. Pay me no mind. :chin:

My understanding was akin to shuffling a deck of cards a lot of times. — tim wood

Yes, I'm not sure what further if any conditions would be placed on that experiment. Combinatoric calculations in probability. Here is a comment from Geology Wiki:

"The Poincaré recurrence time of certain systems is the time for them to revert to a state almost identical to their current state. The system should satisfy the following properties:

1. All the particles in the system are bound to a finite volume.
2. The system has a finite number of possible states.

The universe might not satisfy these properties."

Mathematical theory of PR is pretty strict. :cool:

My contribution here is to refer to "Poincare recurrence." Easy enough to google. Very broad strokes: the idea is that in any system, wait long enough and some configuration of it will recur — tim wood

Too broad, IMO. This is a result that requires a function that takes points in the space under consideration back into that space. However, with regard to a measure defined on the space, the function must preserve that measure. This is quite restrictive. If I examine systems in the complex plane, using the normal Euclidean measure, the function seems to be a simple linear translation. Extremely restrictive.

As a (secular) retired mathematician, I think it prudent to keep an open mind on the subject of a non-religious notion of "intelligent design". Max Tegmark's Mathematical Universe Hypothesis could be so described. The question that arises is, Can there be ID without a specific designer? Or is that an oxymoron?

"Finding the Scorpion submarine:

On May 22 1968 the US navy lost one of its submarines and wanted to find the wreckage, but the intelligence it had was not able to provide an area that was small enough to effectively search. John Craven a naval officer, decided to harness the wisdom of crowds.

He asked a wide group of individuals, drawn from diverse backgrounds ranging from mathematicians to salvage experts to guess the submarine’s location. The group’s average guess was just 220 yards from the location where the Scorpion was eventually found."

theconversation.com 02/09/2016

Honestly, in my 83 years — Frank Apisa

We're exactly the same age, Frank. :smile:

I suspect that the Trump Effect will subside, and I don't fear the man and his policies, but who can tell what will come next? Will the two major Parties continue the fight each other rather than do the peoples' business? I like Uncle Bernie and have long been an advocate of universal health care and easy access to higher education, and were he to ascend to the throne we might move slowly in that direction.

Beg to differ. Functional analysis uses the Hahn-Banach theorem, which is equivalent to a weak form of the axiom of choice — fishfry

Yes, the Zermelo well-ordering theorem comes into play (I took a year of FA fifty years ago), but is not required if the underlying normed linear space is separable (common). I never had any reason to use Hahn-Banach since I didn't do much in soft analysis, but recently I've been intrigued by functional integration and particularly Feynman's path integral (which is not exactly kosher). I dabbled in a generalization of analytic continued fraction theory, so called by an old friend, Wolf Thron RIP. Nothing there hinging upon transfinite set theory (well, to my knowledge!).

jgill you have a university affiliation by any chance? — fishfry

No more, unfortunately. Retired twenty years ago.

You have admirable patience. Thank goodness run-of-the-mill mathematics avoids all this. :cool:

you should have been around through the 1960s. — jgill

I WAS around during the 1960's. The 60's were a blast... — Frank Apisa

Sorry, Frank. I meant the generic "you".

Well, I've not come across your opinions of welfare and UBI before. Interesting to say the least. You should start a thread about that.

I missed direct and intense involvement in the major issues of the 1960s, beginning with 1960-62 when B-52s flew 24/7 from my base, loaded to the gills with nukes, flying out over the Laurentian Shield to the edge of the Soviet Union, then back. Watching the assassination of Lee Harvey Oswald on live TV after the JFK assassination. And of course assassinations of MLK and Robert Kennedy. Huntley/Brinkley with the daily body count from Vietnam (58,318 total American deaths). Dead students at Kent State (1970), and disappearances of civil right workers in Mississippi and Alabama (where I lived through 1964).

But I did attend a Joan Baez concert at Stillman College and sang Kumbaya with her and the chorus. And I visited several hippie communes and was impressed with the industriousness and sharing among the participants.

All in all, I do not consider the 1960s the best of times. The 1960s vs the 2010s? Which era was worst?

↪jgill
I'm not buyin. Please do not contact me with unsolicited services. — Mac

??? I hope this is a joke, and that you have not received a fake message from someone who has hacked my info! :gasp:

I am a vet...served in SAC during the 1950's. We were major league bad-asses...the most bad-ass military force ever on planet Earth at that time. — Frank Apisa

So did I. ADC & SAC 1958-62. Never saw Lemay - he was out of the SAC by then - but an acolyte of his, a one-star, came up on occasion to Glasgow AFB, and he was a nasty SOB.

I have mixed feelings about universal basic incomes and welfare in general. On one hand there are certainly those who need the help, but there are always the scum who take advantage of the system. When I was in England in 1985 to give a talk to the British Mountaineering Council I met a number of healthy and athletic young men who were on the dole, pooling their funds to rent a house, then spending most of their time happily rock climbing, supported by the English taxpayers. And, as a college instructor, I saw how some veterans would get the GI Bill, enroll in classes, then drop out after the first week - happy to get an "F" and then re-enroll the next semester. (After learning of this I would remove them from my rolls early in the term and notify the registrar)

There will always be those who game the system.

As for America's military might, I'd rather it exist than not. I suppose I just don't see the lust. Trump will serve his term(s), then return to his businesses. If you think it has been a trying time since the 2016 election, you should have been around through the 1960s.

Oh boy, here we go again. Might want to reconsider; it's theorems not axioms that could possibly be true, but not provable. :roll: — jgill

Well, I tend to think that axioms can be theoretical in nature? Have you ever encountered such a sentiment in your line of work? — Wallows

To be more specific, it's certain mathematical statements that could be undecidable, including theorems. There are a few examples in combinatorics and number theory, but I'm not aware of such things occurring in complex analysis, for example. If you come across something let me know. As for axioms themselves, I don't think Godel applies. Again, let me know if you come across a contrary opinion.

In the actual work of mathematical research outside of set theory it seems very rare that one might encounter incompleteness. But I am familiar only with my own interests. :cool:

Some of these axioms will remain unprovable as long as the theory is incomplete. — Wallows

Oh boy, here we go again. Might want to reconsider; it's theorems not axioms that could possibly be true, but not provable. :roll:

On another forum, since expired, "free will" attracted quite a bit of attention. There seemed to be a minor consensus that, although scientific studies have implied many decisions are made subconsciously, we, as humans, have agency. That was part of a thread that received over 20K posts. :chin:

Are you aware you have written several paragraphs twice?

It’s mind blowing. — Samuele

Yesterday I thought of a particular condition — Samuele

If we talk about a sense like sight, — Samuele

Was this a test of perception? :cool: