So this is contradictory then is it not? — Philosopher19

One can deduce a contradiction from the assertion that a Riemannian manifold can have a square circle drawn upon it. My point is that it is not a contradiction in terms. Not all contradictions are contradictions in terms. Do you understand Kant's distinction between analytic and synthetic truths? Under Kant's approach, it is a synthetic, not an analytic truth that a Riemannian manifold cannot contain a square circle, just as it is a synthetic, not an analytic truth that 5+7=12 or that 5+7 <> 13.

If something is a square, how can it be a circle at the same time? How can this be imagined in any way whatsoever?

I can imagine it. You need to erase your mental image of a square and a circle, which contains many more properties than are stated in their bare definitions, and focus only on the bare definitions. It might help to start visualising what triangles on the surfaces of spheres look like. They have sides that from some perspectives look curved, and the sum of the angles is greater than 180 degrees. This is admittedly, quite difficult for those that have not worked with n on-Euclidean geometry before.

Yes, only somebody that does not understand mathematics or logic could believe there are contradictions. It is a matter of personal taste whether one accepts it, a bit like the axiom of choice. But as you say, the cost of rejecting the Axiom of Infinity is very high, much higher than rejecting the axiom of choice.

That a set could have an infinite cardinality is what I dispute, as contradictory. "Infinite cardinality" contradicts the definition of "set" as a "well-defined" collection. To be "well-defined" in this mathematical context, of a "set", is to have a definite cardinality, and "infinite" means indefinite.

That is not the mathematical definition of a set. The mathematical definition of a set is that it obeys the axioms of the set theory in which we are working. The most commonly-used set of axioms is Zermelo-Frankel - ZF. The concept of 'collection' does not form part of those axioms.

But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations). We say that a function is 'well-defined' if, using the definition to apply it to an element of its domain, there is a unique object that is the image of that element under that function. The notion of being a set, or of having finite cardinality, is a property, not a function, so the notion of 'well-defined' is not relevant.

If you really dislike the concept of infinity, all you need do is reject the 'Axiom of Infinity', which asserts the existence of a set that can be thought of as the set of natural numbers. Without such an axiom, we can have natural numbers as large as we wish, but there is no such thing as the set of all natural numbers. Such an approach to mathematics is consistent, and some people try to limit themselves to that. The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. Without it, there is an enormous volume of important results that we would not be able to us.

Because you can never have something that is meaningful but can never exist. Can you think of something that is meaningful but can never exist? — Philosopher19

A square circle. It is not, as some think, a contradiction in terms because a circle is defined as the set of all points on a two-dimensional surface that are equidistant from a point called the 'centre', and a square is defined as a shape on a two-dimensional surface that consists of four 'straight' (geodesic) paths that meet at right angles.

It can be proven that no such shape can exist but, because it is not a contradiction in terms (in Kant's terminology, the fact that it cannot exist is not an analytic* truth), one can imagine it existing, and what one imagines is meaningful.

* Earlier I wrote 'a priori' here, which was the wrong term, and refers to something else. I have corrected this now to 'analytic'.

Somebody above (I forget who) said mathematics is not clear about what 'infinite' or 'infinity' means.

In a sense that's true. The symbol:
$\infty$
which is usually read out loud as 'infinity', has different meanings in different contexts. For example it means something different in the expression:
$\sum_{j=1}^\infty 2^{-j}$
from what it means in the statement:
$\lim_{x\to 0}\frac1{x^2} = \infty$

Both meanings, in context, are very precise and formal. But they need the context to know which meaning is intended.

The word 'infinite' means something different from 'infinity', as one'd expect since the former is an adjective and the latter is a noun.

The word 'infinite' is usually only applied to a set, to refer to its cardinality (although it can also be applied to ordinals, but let's not complicate things by worrying about them).

There are two completely different definitions of 'infinite' when used as a property of a set:

1. A set is finite if there exists a bijection between it and a natural number. A set is infinite if it is not finite.

2. A set is infinite if there exists a bijection between it and a proper subset of itself.

As I recall, it is a common exercise in introductory courses in topology or set theory to show that these two definitions are logically equivalent. The Axiom of Choice may or may not be required. I do not recall.

The clean way to do it would be something like lim(1 + 1/2 + 1/4 + 1/8 + ...) = 2. — Magnus Anderson

To be more precise, because the use of ellipsis can sometimes create ambiguity:
$\sum_{j=0}^\infty 2^{-j}=2$
or, for the benefit of those that have an allergic reaction to the mention of infinity:
$\forall \varepsilon\gt0,\ \exists N\in\mathbb N\textrm{ such that }(n\ge N)\Rightarrow \left(\left|\sum_{j=0}^n 2^{-j} - 2\right|\lt\varepsilon\right)$
which doesn't mention infinity at all.

Well that's a definition of 'actual infinite' that is clear and understandable. One can actually work with a definition like that. I suspect however, that that is not the definition that would be accepted by Aristotelians, given my past experience of what they say about it, and of what it says at the link posted above by Devans. That link doesn't use the word 'bounded' at all and is full of woolly, undefined words: 'given, actual, completed'. I particularly like the middle one - an infinity is actual if it is actual. Ah, I see!

Wasn't it recognised several pages earlier that those insisting that there is a clear distinction between the terms 'Actual Infinity' and 'Potential Infinity' are Aristotelians, while the rest are not? Is there any hope of ever coming to a common understanding between Aristotelians and non-Aristotelians, given the fundamentals of their worldviews are so completely different?

I know that the professional mathematicians do not define sets in a way which assumes they must be finite collections. — MindForged

The word "Collection" has an important role in mathematical history because it, along with the alternative "class", was proposed as a name for a group of things (NB the folk, not algebraic meaning of group is intended here) that may or may not be a set. Thus a set is a special sort of collection or class, that obeys certain properties as laid out in the Zermelo-Frankel axioms, or the axioms of some other consistent formulation of set-theory.

In that sense, all sets are collections, but not all collections are sets. So a collection certainly need not be finite.

HOWEVER......

If we approach the issue etymologically, we get a different answer. A collection refers to that which has been collected in the past. So unless we want to postulate an infinite past (which I have no problem with, but your other interlocutors do) a collection, sticking to its etymological roots, must be finite.

For that reason, I think 'class' is a better word to use either as a synonym for set, or for a concept such that all sets are classes, but not all classes are sets. For 'class', neither the etymological nor the history-of-mathematics approach implies, however faintly, that it must be finite.

PS: I don't think it works to give in and let 'set' refer only to finite classes, and make up another word for professional mathematicians to use, because school kids use infinite sets all the time, and would not want to call them schmets. I would not be at all surprised to see questions like the following in a high-school maths exam:

Q1.Tick whichever of the following are true statements about the relationship of the set of even numbers to the set of whole numbers?
A. It is a proper subset
B. It is a subset
C. It is a superset
D. It is a proper superset

Q2. How would you describe the intersection of the set of all even numbers with the set of all multiples of three, without using the words 'even', 'two' or 'three' in any form?

Q3. How many elements are in the set that is the intersection of the set of all integers greater than -2 with the set of all integers less than or equal to 2?
A. 0
B. 2
C. 4
D. 5
E. Infinitely many

All the sets considered here, except for the answer to the third one, are infinite, and high-school kids would have no problem with that.

"infinite set" which is very obviously contradictory.

It can't be all that obvious, since so many mathematicians and scientists have failed to observe the contradiction, and some of them have been reputed to be quite bright.

We must all be grateful that this thread has finally come to light, so that the said mathematicians and scientists can be freed from the delusion under which they have been labouring.

[addition and multiplication of transfinite cardinals is] technically not the exact same operation [as addition and multiplication of the natural numbers] — MindForged

Interestingly, addition and multiplication of real numbers, of rational numbers, and of integers, are also all different from the addition and multiplication of integers:

Starting with the natural numbers, every time we enlarge the set of numbers, the algebraic properties change. There's no reason for us to be surprised when it changes yet again when we move from the reals to the cardinals (including transfinite cardinals).

FWIW, the cardinals form a commutative monoid under addition and a commutative monoid under multiplication, and multiplication is distributive with respect to addition. Like all other sets of numbers, the set of cardinals is totally ordered.

So... I'll leave you to it. — MindForged

A wise decision!

What about pre-writing? Are spoken proper names countable? — Michael

Better to focus on audible names, rather than spoken names, in order to transcend the limitations of the human larynx.

Audible names need not be countable. We could generate an uncountable set of names as follows. Let every name be a sound of length two seconds, that is a pure tone (sine wave) of frequency 800Hz and constant amplitude. We could map the written symbol that is the square whose black part has a height to width ratio r, to a tone with amplitude A + r (B - A), where A and B are widely different amplitudes that are both within the comfortable range of hearing of most humans. Then we distinguish sound symbols by amplitude, and we have an uncountable set of amplitudes from which to choose - the numbers in the interval (0,1).

But why make a commitment when something is inconclusive? — InfiniteZero

Ah, yes - the mantra of commitment-phobes around the world, and fodder for countless movies about indecisive singles driving their would-be-spouse spare with their inability to commit.

Fortunately for those that wish the human species to continue (pax, anti-natalists, I'm not committing either way on that question) people do make commitments when things are inconclusive. They do it al the time.

It’s a problem I agree but I can think of a way past 2 above: imagine as you get closer to the edge of the universe time slows down and right at the edge time stops. So it’s impossible to poke a spear through the edge of the universe because there is no space time in which to poke the spear. — Devans99

That's an interesting idea. If I'm reading you correctly, you're suggesting that there is some point in the universe, call it C (for centre), such that, as we approach a certain number of km from C, we find our movements increasingly constrained and, as we continue, increasingly slowly, we asymptotically become paralysed. It's like there's some kind of sticky force field in the universe that grows stronger and stickier as we move away from C.

It sounds like a great premise for a fantasy novel, a bit like the waterfall at the rim of Terry Pratchett's Discworld. There's no logical reason why it could not be the case.

For me, I find Occam's Razor demands that I prefer an unbounded or hyperspherical universe to this, as they are both much simpler. They can be explained in terms of science we already know, whereas the sticky force-field universe relies on the existence of some sticky force that we have never observed, are unable to test for, and have no reason to believe exists.

So I concede that a finite, non-hyperspherical universe model doesn't have to run into Aristotle's poke-a-spear challenge. But it does require taking on a whole bunch of extra metaphysical hypotheses. I suppose it depends on how determined one is to not have any actual infinities, as to whether that seems attractive.

I never look up links referenced during a discussion. This is for two reasons: 1. Any nonsense can be posted on the web, and often is. 2. If someone has an argument to make, then they should be able to state it in their own words. — LD Saunders

Amen, comrade!

The only exception that I find worth making is when the link is not to an argument but to statistics that are hosted on the site of a credible, impartial authority, that are relevant to the discussion.

Yes; so an object with no start is a non-existent object; — Devans99

That is a misuse of the word 'so'. The word is used after a deduction has been presented, to state the result of the deduction. It is invalid to use it to just state a new assertion that bears no relation to previous assertions, which is what has happened here.

If you consider what you've done here you'll discover that you are using a hidden axiom, which is 'Everything must have a beginning'. Only if we accept that axiom can we deduce your assertion. But accepting the axiom is a matter of taste and I find it completely unintuitive, as well as lacking in any aesthetic appeal, so I don't accept it.

The being could be on any integer if they have counted all the negative integers up to the one they are now on.

The usual objection to that is to ask - 'but what number did they start on?' to which the answer is 'they didn't start'.

The fact that such things are hard to imagine is no reason to suppose that they could not be the case. If there is such a thing as 'the way the world really is' I very much doubt it is something that could make any sense to we cognitively-limited beings.

what are your thoughts on the matter? Can one justify choosing a religion?

It is as easy to justify choosing a religion as it is to justify choosing a spouse, a football team, a place to live, a political philosophy or a job.

Choosing a religion is only problematic if that religion claims that all other religions are wrong. Some branches of Christianity and Islam make that claim, so those branches have that problem. But most religions don't make that claim, so they have no such problem.

Are you aware that denying the actual infinite involves committing to one or the other of the following two propositions?

1. If we travelled far enough through the universe in a straight line we'd end up back where we started

2. The universe has a boundary. In that case, as Aristotle asked, what happens if we go to the boundary and poke a spear through it?

Personally, I find an infinite universe more plausible than either of those.

I see. With the references to 'potential' and 'actual', I see what Sophisticat meant about your view appearing to be based in an Aristotelian metaphysical framework. Like Sophisticat I do not find that framework helpful, so I'm afraid I'll have to bow out.

Nothing can act that is not operational

I am not familiar with that proposition. What does it mean? And why do you feel the absence of an unrestricted PSR is inconsistent with it?

I don't disagree, but I still can't see any support for the idea that a view of the world that does not incorporate an unrestricted PSR would be logically inconsistent.

Authors are not able to compete with the directors and actors in shaping people minds, regardless of the authenticity of their thoughts. — Number2018

I agree, and that is in line with your OP. However my comment about the position of writers was in response not to the OP but to this post that quotes Will Self, which was not about novelists being the most influential people - I doubt they were ever that - but about their being seen as the highest and deepest artists. If we are talking about power to shape people's minds then neither novelists, directors nor actors have anywhere near as much of that power as advertising executives, populist politicians and their spin doctors. But I don't know anybody that views them as the repository of high culture.

I like your reference to Wagnerian Gesamtkunstwerk. For all that novels only reach a minority of the population, and perhaps a smaller proportion now than it was forty years ago, I don't think any medium has replaced it as the closest in people's minds to that ideal.

If the comment was not made in the context of a formal system, what was the meaning of the statement that the world would be inconsistent without an unrestricted PSR, as per the quote below?

Logical consistency — Dfpolis

I think that may still be the case. There are still plenty of Writer's Festivals around the world, where lots of people turn up just to hear authors talk about their work, their views on life, the universe and everything, and maybe read from their books.

Despite there being much more money in Cinema, TV and gaming, we don't see Directors' Festivals or Actors' Festivals. Sure we have the Oscars, but nobody would ever accuse them of being Deep. Nobody expects Quentin Tarantino or Ryan Gosling to have anything particularly interesting to say about the world, but they do expect that of JM Coetzee and Hilary Mantel. Furthermore, the directors and actors are so carefully stage-managed by their media minders that there is scarcely any opportunity to get an authentic thought about the world out of them publicly anyway.

Surely PSR is about completeness, not consistency. Removing axioms from a consistent system cannot make it inconsistent. So if a system including PSR as an axiom is consistent then so too will the system obtained by removing PSR.

But if completeness is what was meant, we hit a block there too, since Godel showed that any system worth bothering with is incomplete.

The chattering classes read because they need fresh fodder to chatter on about. — Bitter Crank

Would I be correct in assuming that this is a humorous, self-deprecating self-reference? Surely, if there is such a thing as a chattering class, there could be no greater epicentre of it than an on-line philosophy forum.

Little did he [Mr Kernan] know that, only seven years later, a publishing phenomenon was about to explode upon the world that eclipsed any literary sensation seen before in any language.

My parents .... believe in hell and damnation, they told their children since they were born that they were fatally flawed sinners that deserved hell (due to the original sin).

I am so sorry to hear that. It must have been really distressing growing up under that influence.

It's probably no consolation to say this, but it may well be that they had those terrible beliefs indoctrinated into them so early in life that they could not resist and could not shake free of them later.

continuous refusal to look at the examples given you after you ask for them — Snakes Alive

I don't recall ignoring any particular example, but it is possible I missed it. Which example do you see as most important to discuss?

I have done both. Unfortunately, it looks like further discussion on this interesting topic will not happen.

I'm sorry to hear that you feel that way. Nevertheless, if you want to put forward an argument on the subject matter itself, I will be happy to engage with it.

In some cases yes. Whether we can do it in an individual case depends on whether we can specify a mechanism. An unsatisfying mechanism that may always work (I haven't thought through whether it would) for cases where the names and the model have the same uncountable cardinality would be to use the axiom of choice, perhaps together with transfinite induction. But same-cardinality cases that we know would work via a non-AC mechanism are easier to grasp:

Consider an alphabet made up of symbols, each of which is a square of side length 1cm that is black below a line at height r cm above the base and white above that, where r can be any real number in the interval (0,1). Then the alphabet has the same uncountable cardinality as the real numbers, and a one-to-one map between the symbols and the reals is that which maps the symbol with line height r to the real number tan((r - 0.5) x pi / 2).

Ask not what is the purpose of religion, but what purpose each individual has in practising their religion. The answers will vary widely between individuals. Common answers would be:

- exploring and expressing spiritual feelings
- using rituals to mark and help cope with important life events
- belonging to a community
- tradition
- providing an efficient and supportive framework within which to help others

Some less attractive reasons that some people have, but would mostly not admit to, are:
- so I can feel superior to others
- so I can wield power over others
- so I can validate my dislike for people from cultures different from my own

In the interests of charity, I assume that most people's reasons belong to the first set. Unfortunately, people motivated by the second set are often prevalent amongst those in power at the top echelons of organised religions.

I don't understand this bit - what do you mean? — Banno

It's just that if one were to try to pick out an individual by printing out the decimal places one by one, one would never be finished picking it out, because at any time there would still be an infinite number of decimal places that had not been printed out yet.

What I am after is, what happens when the model is uncountable? — Banno

An uncountable model of a countable theory will contain uncountably many elements that cannot be individually picked out by any constant term. But the L-S theorem tells us that the model will contain a countable sub-model. A countable submodel of the real numbers, that satisfies all the axioms of the real numbers, can be constructed as follows:

1. Let N be the natural numbers.

2. Define map f from the power set of the reals to itself such that, for any subset A of the reals, f(A) is the set of all reals that can be picked out as the unique satisfier of a proposition with a single free variable that only uses symbols from the theory's alphabet and real numbers in A.

3. Define D to be the transitive closure of N under the operation f, that is:
D = N ∪ f(N) ∪ f(f(N)) ∪ ........

Then D is a countable subset of R that satisfies the axioms of the real numbers.

This is an application of the downward L-S theorem. The upwards L-S theorem says that there are also models at all cardinalities larger than the one we started with. But that is less surprising (to me) because it just involves tipping in lots more unnamable individuals.

No.

No

What if one used an uncountable alphabet? — Banno

Then we could refer to an uncountable number of individuals using sequences of just one letter.

Whether that entails that we can use it to refer to all the real numbers depends on whether we accept the Continuum Hypothesis. Indeed the continuum hypothesis is precisely the assertion that no uncountable set has a lower cardinality than that of the reals. It can neither be proven nor disproven by ZFC - the foundational axioms of most of mathematics.

No.