A heap is more than one, but it's not exact. — ssu

A heap is an estimated size - looking at the heap is an act of rough measurement of size. That is possible only with finite objects; you cannot estimate the size of infinity; so I maintain infinity is unmeasurable so has no size.

Meaning that they do have a size, but the size is obviously unmeasurable — ssu

If the size is unmeasurable then it is not a size. Size has to be an integer.

So, could it be then there would be Absolute Infinity? — ssu

In our minds yes but it can't exist as a consistent mathematical object:

∞+1=∞
implies
1=0

And it can't exist in the real world:

- Creating anything infinity large is impossible; not enough time / would never finish
- Creating anything infinity small is impossible; no matter how small it is made, it could still be smaller
- Spacetime is a creation so nothing in it is infinite

But isn't an infinite set bigger than any finite set? Doesn't that imply size, even if obviously you cannot measure it like a finite set? — ssu

An infinite set is unmeasurably bigger than a finite set. An infinite set therefore has no size.

Or: size is an integer property. Infinity is not an integer, if it were, it would be an integer X greater than all other integers. But X+1>X. So infinity is no sort of size or number.

Yeah, you cannot just say that set theory is wrong. You would actually need to give a proof of it in mathematics. — ssu

I'll define infinity as ‘A number bigger than any other number’ then it is clear that there can be only one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity.

That disproves the assertion of multiple infinities in set theory. From that we can deduce that:

∞+1=∞
implies
1=0

IE infinity is not a number... disproving the transfinites from set theory.

God's existence wouldn't make life "objectively significant" because significance is always subjective — Relativist

If it's subjective for God, its objective for everyone else.

The only fact in evidence is: there is life in the universe. But this fact is consistent with both alternatives, so it doesn't make your preferred alternative any more likely. — Relativist

But the two alternatives are not equally likely:

1. By chance is a billion to one
2. By design is chance of God existing (say 10%) * chance of God being interested in life (say 10%) giving a hundred to one

So the chance purposeful fine tuning are 10 million times higher than the chance of a fluke.

Well.. I don't like it too, but nobody has shown that is inconsistent yet, and it's used since a very long time. So, I would guess that it's not inconsistent! — Mephist

Maybe I'm using inconsistent in the wrong way; set theory is not inconsistent within itself, but to me, Galileo's paradox and the other results bijection give are inconsistent with common sense. Especially this inconsistency with logic is noticeable for transfinite arithmetic.

Sorry, I don't know transfinite aritmetics.. :sad: But if you have some good links to documents that explain what is it I would be interested! — Mephist

This document is reasonable (see page 23):

http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Yarnelle%20-%20Transfinite%20Math.pdf

Naught and c; whether c equals 1 is an open question. And nothing else? That's just plain wrong. — tim wood

Well how come nearly everything gets classified as aleph-naugh:

the set of all square numbers, the set of all cubic numbers, the set of all fourth powers, ...
the set of all perfect powers, the set of all prime powers,
the set of all even numbers, the set of all odd numbers,
the set of all prime numbers, the set of all composite numbers,
the set of all integers,
the set of all rational numbers,
the set of all constructible numbers (in the geometric sense),
the set of all algebraic numbers,
the set of all computable numbers,
the set of all definable numbers,
the set of all binary strings of finite length, and
the set of all finite subsets of any given countably infinite set.

Then there is aleph-one the continuum. No-one has ever found any other aleph number (that was not generated from a previous aleph).

What use is it a measurement when it is so crude a measurement?

Sorry, using 2^S to denote the power set of S. The proof I gave is meant to show that the set of all sets does not exist. I maintain that it is the cardinality of the set of all sets that does not exist.

Well, the problem is to give a definition of "size" for sets that you cannot count — Mephist

I don't see the need to assign a size to something that is unmeasurable - it leads to nothing useful, just paradoxes.

I think it would have been better if it had been recognised by Cantor that there are two different sorts of set - finite and infinite. An infinite set is not a-kind-of finite set and vice-versa; as can be seen by the two object types having very different properties:

- A finite set has a completely defined list of members. An infinite set does not have this property

- An infinite set does not have a cardinality property: cardinality or size implies the ability to measure something. Infinity is by definition unmeasurable so infinite sets have no cardinality/size property

Instead Cantor just made up fictitious numbers (transfinites) for the non-existence cardinality property of infinite sets. I'm an ex-computer programmer and what Cantor did is regarded in software as a cardinal sin (pardon the pun).

A finite set can be formed in reality. Infinite sets are not part of reality and are just a conceptual thinking aid. I think maths should not try to treat the two the same.

Saying "it is only natural" does not constitute objective evidence. Consider that in a fair lottery, some random person will win - and yet (per your admission) every winner will suspect the lottery was rigged for him, but he will be wrong — Relativist

We don't know if its a fixed lottery or not - we know nothing about it so we have to assume a high probability that it is rigged.

- If its a fair lottery we have 1 in a billion chance
- If its a rigged lottery we have 100% chance

If you really knew nothing about the lottery and you won, you'd have to conclude it was very probably rigged.

To be honest, I think you've hit on the source of the fallacious thinking: it's based on the unsupported belief that there "must be a reason" for all subjectively significant events. — Relativist

Life is objectively significant if you factor in the possible existence of God (who would want intelligent life).

We might be going around in circles here. Maybe we just have to agree to disagree on this one?

On the number line between zero and one: are there more rational than real numbers? Or the other way round? To know, you have to have a way of quantifying both. How do you go about these? — tim wood

I would not use bijection to arrive a conclusion about a sets size; bijection claims that there are the same number of rationals as naturals so it is clearly wrong (naturals are a proper subset of the rationals so bijection is giving a wrong results).

I would recognise that infinity has no size so is not measurable.

I would recognise that what Cantor was trying to do was to assess the size of infinity (which is impossible). Instead I'd look to compare the density of rationals and reals on the real number line. Bijection is one possible way of measuring density but its frankly pretty useless: we have aleph-naught and aleph-one but there is nothing else!

That's correct, but my point is that the mere fact that the dwarf won does not serve as evidence that the organiser wanted the dwarf to win. It's POSSIBLE that he did, but there's no basis for considering it probable — Relativist

I would say if you only enter one lottery in your life and you know nothing about lotteries except it is a billion to one shot and you win, then it is only natural to suspect the lottery was rigged.

So there is only one universe (=only one lottery) and we know nothing about what kind of lottery it is except:

- Its a billion to one shot
- There is a non-zero probability of a God (who would want us to win the lottery)

The probability of a God existing is greater than the probability of winning the lottery / getting a life supporting universe by sheer luck and God would want a life supporting universe.

So we can conclude the chances God fine tuned the universe for life are much higher than its life supporting by a fluke. So we should go with the most probable explanation.

Agreed, actual infinity is a usual mental aid.

The problems can up start once you try to do anything logical with it. It's not a logical concept so it leads to paradoxes. Particularly if you insist (like set theory does) that infinity is measurable and has a size then it leads to paradoxes.

It is also a problem that the inclusion of actual infinity in set theory legitimatises actual infinity and then cosmologists and other scientists come along and build actually infinite models because they think actual infinity has a sound basis in logic when it does not.

- I think potential infinity (calculus) is obviously very useful in maths and science
- I think actual infinity has no useful applications

God really wanted some intelligent life to entertain him so he created some (14 billion years into the whole project, for some reason), stuck it one one tiny planet in the middle of empty space, populated the rest of the entire universe with lifeless rocks and then buggered off (minus one brief showing to set fire to a bush and hand down some instructions about neighbour's oxen). — Isaac

1. God is timeless so 14 billion years in not relevant
2. The whole universe is teeming with life. Most star systems will turn out life supporting like ours.
3. There are 10^23 star systems in the observable universe. By comparison, there are only 10^10 years since the Big Bang. That means God could not have found us even if he wanted to - there is plenty of life to choose from in the universe so why would he be looking for us? There is no reason we should have 'heard' from God.

This isn't news to mathematicians. When the concept of infinity was invented, there was a (perceived) need for it to be integrated into mathematics. (The alternative was to leave infinity standing alone and lonely, and this (apparently) was unacceptable.) The mess you observe is the result of that 'integration'. — Pattern-chaser

That's a realistic attitude, but my personal experience is it is not shared by many mathematicians. They tend to get very defensive whenever infinity is questioned.

1. Creating anything infinity large is impossible; would never finish
2. Creating anything infinity small is impossible; no matter how small it is made, it could still be smaller
3. Space cannot be infinite because it is expanding
4. Time cannot be infinite because an infinite temporal regress is impossible
5. Only in our minds can things continue ‘forever’; in reality this would be akin to magic

If we can establish that actual infinity is not a number and is not part of our universe, there will be an opportunity to return to set theory and clean up the mess.

You have avoided responding to my analogy of a one-eyed, hemophiliac dwarf (I'll abbreviate as OEHD) winning a lottery — Relativist

My point on the dwarf is that his unique set of characteristics happen to correspond to what the lottery organiser wants to win the lottery (is the closest analogy I can think of).

Or that the unique set of characteristics required for a universe to be life creating happen to correspond to what God would want from a universe - for it to be live supporting.

Regarding your case for a creator - Try again. You didn't show that the creator would be likely to desire life to exist. — Relativist

If you were God, the first cause, would you not be lonely/bored? If you had the means and opportunity, maybe you'd create something to keep you occupied. Inanimate matter is boring, so something living. On the basis that life is good, you'd make it as large as possible (larger=more good). IE you'd create a life supporting universe.

And then you have to be careful, which you neither are nor appreciate the need to be, with the concepts applied to transfinite sets. That is, basic arithmetic functions don't work quite the same way. And so on. — tim wood

Basic arithmetic does not work at all with the transfinites:

∞+1=∞ implies there exists something that when changed, does not change
∞/2=∞ falls foul of the axiom 'the whole is greater than the parts'

(both cases flaunt logic)

Regarding the epistemic probability of God: bear in mind that you're referring to specifically to a God that would choose to create life. What evidence is there for such a God? Cosmological arguments only point to a first cause; contingency arguments only point to there being a creative force that exists out of metaphysical necessity. What objective basis do you propose for assigning a probability to a God that wants to create life? — Relativist

1. Causality absolutely requires a first cause (else nothing would exist)
2. The first cause must be able to effect something without in itself being effected
3. So the first cause must be self-driven, IE intelligent
4. An intelligent first cause would want a universe with intelligent creatures in it.

So on the basis of the above argument, I can assign a non-zero probability to God's existence which is much higher than the chances of the universe being life supporting by accident.

OK whats wrong with this proof?

Infinity is not a number:

if infinity was a number, it would be a number X greater than all other numbers. But X+1>X

Clearly you know how many even numbers there are and you know how many natural numbers there are. And since you argue the set of natural number is the larger, then in consequence it must be possible to count the even numbers. Please do so and tell us how many even numbers there are. — tim wood

If you read any of the above, you would have gathered that I maintain infinity does not have a size so it is impossible to measure the size of infinite sets.

All we can do is induction - for any reasonably sized interval, there are more natural numbers than even numbers.

Back to my analogy, a one-eyed hemophiliac dwarf winning is consistent with a lottery that is rigged for one-eyed hemophiliac dwarves, and also consistent with an honest lottery in which everyone has an equal chance. You're treating the mere fact that such a person won as evidence of a dishonest lottery. — Relativist

If you only entered one lottery in you life and you won at a billion to 1, would you not find it suspicious? This is the situation with the universe; there was only one lottery for life supporting attributes, our universe won the jackpot; it seems highly suspicious.

If you do not treat life as a design objective, there is no relevant coincidence. — Relativist

But God independently of fine tuning has a non-zero chance to exist. So there is a non-zero chance of a design objective which dwarves the chance of a fine tuned universe happening by accident.

"the set B is bigger than the set A if there isn't any function that for every element of A gives an element of B and covers all B" ( i.e. each element of B corresponds to some element of A ) — Mephist

This is a nonsensical definition: for instance, it claims the even numbers are the same size as the natural numbers (as there is a one-to-one correspondence between the two). But the even numbers are a proper subset of the natural numbers. If either had a size, the size of the natural numbers must be greater than the size of the even numbers.

This is what it means "an infinite hierarchy of of infinite sets each one bigger than the other". — Mephist

If you use a reasonable definition of infinity: ‘A number bigger than any other number’ then it is clear that there could only be one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity.

In the standard contemporary mathematics based on set theory you can't speak of the "set of all sets" because it's not a set itself — Mephist

The set of all sets does not exist! Standard ‘Proof’:

1.Let S be the set of all sets, then |S| < |2^S|
2. But 2^S is a subset of S, because every set in 2^S is in S.
3. Therefore |S|>=|2^S|
4. A contradiction, therefore the set of all sets does not exist.

What is wrong with this ‘proof’?

- It is that the cardinality of the set of sets does not exist (infinite sets do not have a cardinality - infinity - is unmeasurable)
- This proof does not prove that the set of sets does not exist. This ‘proof’ is a sham
- Once it is acknowledged that the cardinality of an infinite set does not exist the whole of infinite set theory collapses like a pack of cards.

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica — Mephist

It is the assumption that infinite sets are measurable that invalidates naive set theory. ZF set theory is patchwork of hacks that tries to cover all the the holes and fails - the solution is to acknowledge infinite sets do not have a cardinality / size.

How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:

∞ + 1 = ∞

This assertion says in english:

’There exists something that when changed, does not change’

Thats a straight contradiction.

You spout ignorance. — tim wood

Point out my instances of ignorance then...

I am listening its just I don't agree with what you are saying.

There is not a 'probability' of the existence of God. You have misunderstood how probability works and you have already had this repeatedly explained to you. — Isaac

So you are 100% certain God does not exist? How did you arrive at such a conclusion? I'm sure the world would like to know... there has been some debate on the issue.

But there is a non-zero probability of the existence of God; IE someone was aiming for precisely that (a life supporting universe).

And the non-zero probability of the existence of God is much larger than a billion to one shot of it happening by a fluke .

So the only possible conclusion is: the universe is probably fine-tuned for life.

When all your 'guesses' point in the same direction?

It doesn't 'happen to be' it is the only number it could ever possibly have been because had it been any other number we wouldn't be here. We are here, so it is completely unsurprising that it is that number. — Isaac

You are not asking the right question. We already know it had to be that number else we'd not be here to talk about it. The question should be 'Why was it that number?'

By massive fluke or by design. A simple choice.

BUT IT HAPPENS. — Frank Apisa

Yes but the question is why did it happen? Has a gigantic fluke come off? Or was it not a fluke at all? Its quite clear to me that the second is vastly more likely.

But that happens to be the number that gives us a life supporting universe. Do you honestly not find that a staggering coincidence?

If there was a God he would certainly want a life supporting universe.

In this case, 2+2 does indeed equal 4 - the universe is fine tuned for life.

Even if you think there is a very low probability of God existing, it is still a much more likely explanation than a billion to one shot coming.

Take the strength of gravity for example; it could have been set to anything between 0 and ∞. It's set precisely so that stars and planets can form and so that nuclear fusion in stars takes place at the correct rate (and we don't get too many black holes).

That is just one of 20 or so constants that all have to be within given ranges. It is much more than a billion to one shot I would guess.

We are here to wonder about it so it is not in the least bit surprising that the universe is supportive of life. In fact, it's an absolute pre-requisite for us being able to ask the question. — Isaac

That the universe must be live supporting is a given; the real question is why is the universe life supporting?

1. A billion to one fluke comes of (that all 20 odd constants came out in the life supporting range)
2. Or the universe was fine tuned to be life supporting

2 is much more like that 1.

You are assuming the fundamental constants could have been different, so each combination of constants participates in the lottery. — Relativist

The fundamental constants could of all been very different:

- The strength of the 4 forces
- The masses and charges of the subatomic particles
- The rate of expansion of the universe

That gives a huge possibility space - and we have only one shot at winning. We won, so it is much more likely it was due to the lottery being rigged (=the universe being fine tuned).

You seem to be blind to the fact that you are treating life as a design objective. If you do not treat life as special, your argument falls apart. If you do treat life as special, your argument is circular. — Relativist

If there was a God (lets say there is a 25% chance of that just for arguments sake), life would be a design objective. So we have:

- 25% chance of God * 100% chance of fine tuning
- 1 in a billion chance that we 'get lucky' and have a life supporting universe without God

Which of the above is a more likely explanation?

Specific counter arguments rather than waffle please.

But in this lottery there is only one participant - only one of the billion tickets was bought - as represented by our universe. So it is remarkable that we won.

You're overlooking that every one of the billion possibilities had an equal chance of being drawn (1 in a billion), and therefore it's not remarkable that the winner was a 1 in a billion shot. — Relativist

But it is still remarkable that we won at a billion to one - there was only one lucky ticket (the one life supporting universe). Suspiciously remarkable. Fine tuning is a much more likely explanation that a billion to one shot coming off.

What he's says there about absolute infinity not being subject to maths... if only he'd realised that was the case for infinity full stop (there is only one infinity).

Agreed. He was mad:

“I have never assumed a ‘Genus Supremum’ of the actual infinite. Quite on the contrary I have rigorously proved that there can be no such ‘Genus Supre- mum’ of the actual infinite. What lies beyond all that is finite and transfinite is not a ‘Genus’; it is the unique, completely individual unity, in which every- thing is, which contains everything, the ‘Absolute’, unfathomable for human intelligence, thus not subject to mathematics, unmeasurable, the ‘ens simplicis- simum,’ the ‘Actus purissimus,’ which is by many called ‘God.’ - Cantor.

He though God was talking to him and giving him these ideas about infinity. It's all marsh gas.

Cantor. The transfinite numbers in set theory.

Proof infinity is not a number

First, we realise if two different types of infinity existed, they would have to be larger than each other. Thats a logical contradiction, so we must of made a wrong assumption; only one kind of infinity can exist.

Then:

∞ + 1 = ∞

Leads to:

1 = 0

Which is another logical contradiction; infinity is not a number.

This is how I see politics (and decision making in general):

Net Pleasure

I use a definition of pleasure and pain includes all forms of pleasure and pain: physical, emotional, spiritual, intellectual etc… I can then define ‘net pleasure’ as:

Net Pleasure = Pleasure - Pain

So for any decision, it is in theory possible to calculate the resultant ‘net pleasure’ for each of the possible outcomes. This is the variable that all intelligent life forms try to/should try to maximise during decision making.

Long Term > Short Term

The key determinant in maximising net pleasure is a consideration of the implications of the decision in the long term. Because long term > short term, net pleasure is maximised by doing what is right in the long term, even if it involves short term pain.

Even for senior citizens, life expectancy is around 10 years, so we are all long term creatures and all benefit from long term decision making.

Right Decisions / Policies

Right is typically ‘what is right in the long term’ - short term pain is exchanged for a larger amount of long term pleasure

Wrong Decisions / Policies

A wrong decision is typically ‘what is right in the short term’: short term pleasure is experienced but is outweighed by long term pain.

Left and Right Wings in Politics

Left is another name for wrong. So we have 'wrong wing parties' and 'right wing parties'. But the left invest more in the long term - which is the right thing to do. So right and wrong are the wrong way around. It should be as follows:

Right Wing / Republican - Short term. Should be called the 'Wrong Party'
Left Wing / Democratic - Long term. Should be called the 'Right Party'