He was indeed inspired by Cantor, but he also achieved some of the same results and reached some of the same conclusions at least semi-independently. In the end, he became disenchanted with Cantor's whole approach; as Rich has been emphasizing, you cannot adequately represent true continuity with something that is discrete. — aletheist

What is discrete in the Reals? What aspect of the Reals is being inadequately represented by this discrete thing?

The continuum was not discovered via set theory, it was (and still is) modeled using set theory. Real numbers merely constitute an analytic continuum; they do not form a true continuum as defined by Peirce - as well as duBois-Reymond, Brentano, Brouwer, and many others. — aletheist

And those later objections have been swept aside. Cantor was the first to rigorously define the continuum in 1870s and all the dissenters have been forgotten.

I think you'll find that Peirce got into the act somewhat later than Cantor, after being inspired by Cantor. And, in the history of Real analysis, set theory, etc, Peirce is a dead-end. Cantor's ideas have been extended and developed, Peirce's have been abandoned.

The problem is that since the late 19th century, mathematics has largely relied on the manipulation of the discrete, because it has been grounded primarily in set theory. In recent decades, category theory has emerged as a viable alternative that is more general and much more compatible with the concept of continuity. — aletheist

What? Have you not been paying attention? The continuum was discovered via set theory!

OOps! Hydropower now proved to cause dangerous global warming. Officially no longer "green".

https://academic.oup.com/bioscience/article/66/11/949/2754271/Greenhouse-Gas-Emissions-from-Reservoir-Water

“The new study confirms that reservoirs are major emitters of methane, a particularly aggressive greenhouse gas,” said Kate Horner, Executive Director of International Rivers, adding that hydropower dams “can no longer be considered a clean and green source of electricity.”

No, the act of dividing something that was continuous causes it to become discontinuous. Not surprisingly, we disagree on whether the infinite divisibility of a line renders it discontinuous, even if it is not actually divided. I am never going to convince you that "x-able" does not entail "actually x-able," and you are never going to convince me that the two are necessarily equivalent; so we might as well just agree not to waste each other's time by going down that road yet again. — aletheist

What does "dividing" even mean? If you mean that you can take an interval of the Reals, well we learned how to do that at school, so we can do it. We can even write down an expression for taking (countably) infinite number of (finite) intervals of the Reals. So it is actually and trivially x-able.

I leave it as an exercise to the reader to ascertain whether the intervals must be open or closed.

The whole modern world economy is a result of inexpensive oil, and there is nothing that can "fill its shoes". — Bitter Crank

There certainly isn't, but don't discount the worlds fastest growing source of energy, which nearly matches oil in terms of energy production - coal.

Natural gas, is also keeping up with oil in terms of growth rate.

The act of dividing something demonstrates that the thing divided is not continuous. — Metaphysician Undercover

Well, you can certainly partition the continuum of the Reals wherever you wish.

Shale oil is not a US innovation. It predates the US by a couple of hundred years. — Benkei

The UK also has vast stores of hydrocarbons that could be fracked - hundreds of years worth, but Scotland prefers to import US fracked gas.

Not in the manner you describe. The idea behind biofuels is to use biomass and catalytically convert it to fuels. the biomass is quickly replanted and regrown and therefore "captures" the CO2 released from burning the biofuels. In essence, nothing more than speeding up the process by which fossil fuels are created naturally. CO2 capture and catalytic conversion to methanol/ethanol is the same principle. — Benkei

I described the facts behind biofuels - increased CO2, forest and habitat destruction, subsidies, but forgot to mention the inevitable increase in food prices. What you describe is the fantasy.

e.g. The UK is the biggest importer of wood pellets in the world. Forests are destroyed in US, the wood chipped and kiln dried, made into pellets, shipped to UK and burned at Drax. All at extreme financial and environmental cost, and at increased CO2 emissions. Trees cannot grow that fast!

So do biofuels. I meant, of course, in a commercially, viable manner. — Benkei

So long as you consider massive subsidies and destruction of primary forest habitats "commercially viable", which it certainly is if you are in receipt of the handouts.

Oh, and if you don't care about the increase in CO2 over simply burning coal.

You're not joking are you?

Nuclear fission would be great if we can get it to work. — Benkei

Nuclear fission works quite well.

No one is disputing that actually dividing a continuum introduces a discontinuity. However, that discontinuity is not there until we break the continuity by that very act of division. — aletheist

Could you give an example of how you "actually" divide a continuum, and "introduce a discontinuity"?

Literally Hitler, and there are many of these:

http://www.independent.co.uk/news/world/americas/adolf-hitler-donald-trump-mein-kampf-bluffed-way-to-power-nazi-leader-germany-fuhrer-us-president-a7568506.html

A summary of the WaPo "Trump is Hitler" meme

http://www.wnd.com/2016/10/5-washington-post-writers-liken-trump-to-hitler/

How could you possibly know that? — Thorongil

We know it for sure, because that's the media narrative. Don't forget, Trump and his non-haters are literally Hitler.

The statement "Torturing children is wrong" is ambiguous. It can mean torturing children is wrong in general, or it can mean torturing children is always wrong. — Baden

If you are looking for a set of rules to impose on others, then perhaps it is ambiguous. However, if you are looking for a moral theory with which to inform our decision making, it is not. Torturing children is wrong, no matter how perverse your trolley problem.

That's my basic difficulty with 'moral realism'. I can't think of a moral-sounding assertion that is factual. — mcdoodle

Do not harm the methods of error correction.

I actually prefer to dismantle reason, over dismantling the world. That talk and theory and reason is in the end inadequate to the world is relatively unproblematic; we can always just shut up about what cannot be said. And that seems preferable to trying to excise it from the world. — unenlightened

Sure, when PROVED wrong, just ignore it.

As such, it is certainly vulnerable to being shown to be contradictory or incoherent, but since there does seem to be a world, and we do talk about it both as a totality and as fragmentary facts, it is so fundamental to discourse that it might well be easier to dismantle set theory if it proves to be in contradiction with such a statement. — unenlightened

You will need to dismantle Relativity as well as Set Theory, not to mention reason if you want to maintain a "totality of facts".

e.g. https://en.wikipedia.org/wiki/Rietdijk%E2%80%93Putnam_argument

I suppose this means that there can be no beginning point of a wave. Such a beginning would be a discrete occurrence. — Metaphysician Undercover

But the particle must be within one of the discrete grooves.

I disagree, but that is the nature of the world. You wanted evidence, I gave it to you. Do what you wish with it, including ignoring it. For those who wish to follow this line of inquiry, they are free to do so — Rich

Your msunderstanding of QM is not evidence for infinite continuous fields, which by-the-way can't be continuous if they are quantized.

It's a rich line of inquiry. Other key ideas are Bohm' quantum potential and how it might explain the delayed choice experiment. — Rich

Sorry to break it to you, but entanglement has nothing to do with continuous fields that extend to infinity.

You've still not shown a contradiction; only that the number of facts is not countable. — Banno

No. I showed that the ASSUMPTION that there is a totality of facts leads to a CONTRADICTION. The argument works whatever Aleph number you assign to the totality.

Quantum entanglement. — Rich

What has entanglement got to do with the existence of continuous fields of infinite extent?

Love the argument, Tom; but I have to say I agree with Pierre-Normand that what you have shown is that the totality of facts is uncountable, not that it is impossible. — Banno

You don't understand Cantor's theorem then. The powerset is always strictly bigger than the set, as Cantor proved. This is how the various Aleph numbers are generated. Even if the set of facts was Aleph_3, its powerset would be Aleph_4.

There is certainly empirical evidence for fields which are continuous and stretch forever. — Rich

Could you point me towards the empirical evidence for these fields which are continuous and stretch forever?

That really depends on the facts of physics. Your proof is a perversion of Zeno's paradox as you state. — Question

Nothing to do with Zeno's paradox. It's taking the power set and Cantor's theorem, as I explained.

And, if supertasks exist, then the CTD-principle is false.

This is clearly not true. A computer is a logical space, which behavior is dictated by logical facts. Ask Turing. And as per the Church-Turing-Deutsch principle, the world is the totality of facts, not things. — Question

But I've just proved there is no such thing as the totality of facts, and while doing so I have respected the CDT-Principle.

You've only shown there to be no infinite and denumerable totality of facts. There could still be a finite, or a non-denumerable, totality of facts. At any rate, that would not be ruled out on the basis of such a proof. — Pierre-Normand

Does Cantor's theorem not work for finite sets? I thought there was a well known relation between the cardinalities of a set and its power set?

As for uncountable sets, I'm pretty sure that Cantor proved that P(T) is always bigger.

I'm not sure if people can look past through the profundity of this statement; but, this is essentially saying another way that the totality of facts is that and only that what an omniscient being can perceive. — Question

Inspired by the recent thread on Zeno's paradox, I shall prove that no "totality of facts" can exist:

Let T be the set of all facts T={t1, t2, t3, ...}

Consider, further all subsets of T, which are the elements of the power set of T, P(T) :

{}
{t1} {t2} {t3} ...
{t1, t2} {t1, t3} ...

To each element of the power set there will correspond a fact, which we construct like this:

t1 $\notin$ {}
t1 $\in$ {t1}
...
t1 $\in$ {t1, t2, t3, t4}
t1 $\notin$ {t2, t3, t4, t5}
...

Of course, there is nothing special about t1, a set of facts can be constructed similarly with any tn.

By constructing our new set of facts, we have a set with as many members as there are in the power set P(T). But, by Cantor's power set theorem, P(T) is always strictly larger than T.

Thus there are more facts than members of T, therefore no "totality of facts" can exist QED.

Moving back toward the original question of this thread, I'm eager to introduce the notion of Supertasks to the conversation. A great summary with examples of Supertasks can be found here — Voyeur

Why is motion a supertask rather than a hypertask?

What exactly is it that you think I am not comprehending? Sincere question, I am eager to learn. — aletheist

I may have been responding to you, but I thought it obvious I was not referring to you!

It is a mistake to confuse mathematics with metaphysics. — aletheist

I think it is a bigger mistake to write off the great works of Georg Cantor and Nicolas Bourbaki (that makes at least 8 geniuses) on the basis of your personal inability to comprehend the first thing about it.

A floozable set is a set with the same cardinality as some subset of the set of natural numbers. — Michael

The set {1,2,3,4,5} has the same cardinality as {6,7,8,9,10}, so we should call it foozable? How do you calculate the cardinality? Do you fooze the set, or do you perhaps count it's members?

No, it's the same definition and both relate to bijection. It's just that the cardinal numbers used are different. In the case of finite sets we use natural numbers and in the case of infinite sets we use aleph numbers. — Michael

It might be worth noting that Cantor proved that any interval of the Reals [a,b] cannot be placed in one-to-one correspondence with the Naturals, before he developed the idea of cardinality.

So, instead of getting bogged down with definitions, the remarkable discovery that there are more Real numbers in any finite interval than all the Naturals, indicates a fundamental difference between the continuous and the discrete.

Of course, this difference was analysed further and more remarkable discoveries were made, but it is the distinction between the continuous and the discrete that is of fundamental importance.

Another remarkable feature of the continuum is that in any interval [a,b] there are an uncountably infinite number of transcendental numbers.

A one-to-one relationship between sets e.g. from the natural numbers onto itself (i.e. a permutation) is not a one-to-one function, but a one-to-one and onto function. An easy slip-up to make when moving from the language of relations to the language of functions, wile being brow-beaten by unreason.

Sorry, I got my terms mixed up. It's surjection, not bijection when it comes to infinite sets. — Michael

Not that it matters one jot due to the level of willful ignorance on display, but I think you were quite correct to use the term "bijection" = one-to-one and onto.

I'm no longer finding it amusing to follow the obtuse denial of well established mathematical truths, so I can't say for sure because I can't be bothered to tease out the remnants of sanity in this thread, but I get the feeling that a surjection will not suffice for your purposes.

I am neither a mathematician nor a philosopher, but that statement seems consistent with the claim that the real numbers do not qualify as a true continuum in Peirce's sense, since they skip over those infinitesimal intervals. — aletheist

Wow! I'm out!

The values ARE equal. There is NO difference between them.

You cannot create a new number by adding an infinitesimal quantity.

It's you.

And by the way, which real number is bigger:

1 or 0.999...

According to you, or Peirce?