Therefore, this soteriological system is unjust — Empedocles

You could widen the argument:

- You could argue that people’s propensity to sin is determined by their nature and nuture
- neither of which they can do anything about
- so it’s unjust to punish people in this world or the next (unless it’s ‘corrective punishment’).

If the universe turns out to be virtual like ‘The Matrix’ then our soul is really just our information. It’s possible to move information between computers so in theory the transmigration of the soul might be possible.

If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions. — SteveKlinko

A good point. It depends on your view of time as to whether you think the particle exists in an actually infinite number of states:

- Presentist. The past does not exist. So the particle does not exist in an Actually Infinite number of states, just one state, the present.
- Eternalist. The past exists so continuous time implies the particle must exist in an actually infinite number of states.

Presentism leads to paradoxes, so that suggests Eternalism. But time must be discrete for Eternalism to be free of Actual Infinity (which I class a paradox).

Calculus has problems too. For example the infinite series 1/2^n

1 + 1/2 + 1/4 + 1/8 ... = 2

Logically it’s incorrect to write =2 should be ~2. It’s only a small error but the sum of that series is always less than 2.

The way you are solving the paradox uses the undefined quantity ‘infinity’ but I acknowledge there are other ways out of Zeno’s paradoxes other than discrete space.

Still I’d argue for discrete spacetime on the grounds:

- there is no such distance as 1/oo mathematically.
- Imagine a particle moving over a finite period of time. Continuous spacetime would require the particle to have occupied a actually infinite number of states which is nonsensical.

Still even if space is continuous, that would only be a potential infinity rather than actual infinity.

There is no largest natural number X because X+1>X.

The natural numbers are defined, but not as a set, just the description of how to populate a set.

The set of natural numbers is undefined.

If we were to get the name of everyone in the world as write it down we would have the set of human faces.

- The concept of a human face can be defined
- The ‘set of all human faces’ is a finite list so in principle is also definable (as a set)
- the description ‘set of all human faces’ is not a complete definition of the set (so is undefined)

Actual Infinity is not a quantity:

- There is no number X such that X > all other numbers
- Because X+1 > X

Space is discrete that’s why we get paradoxes when we assume it’s continuous (Zeno’s paradoxes).

Unless you can argue that the notion of a one-to-one correspondence is logically incoherent — MindForged

Well one-one correspondence is logically flawed: There are the same number of natural numbers as square numbers? Surely a paradox - a sign we are dealing with a logically flawed concept.

We are comparing two undefined things and we get nonsense.

I, for instance, do not find the beginning of time to be any easier to conceive than an infinite past, — SophistiCat

An in infinite past leads to logical contradictions so time must have a start:

- the measure problem. Everything that can happen will happen, an infinite number of times
- this breaks probability; everything becomes equally likely
- Reductio ad absurdum, time is finate and has a start

We can use reason to determine that we have no methodology proven to be capable of analyzing anything the scale of gods — Jake

I think you can analyse god. For example, if you are a materialist, you must deny the Actually Infinite exists, so god is finite.

The set of rationals is easily defined — tim wood

How do you completely define something that is larger than any given finite number:

- You lack infinite paper to write out a definition
- You lack infinite mental power to visualise an infinite set
- R or ‘the set of rationals’ is merely the selection criteria for the set not a full description of the set itself
- Actually Infinite sets are not fully describable so are NOT DEFINED

They don't produce two different concepts. — MindForged

But they do: the ‘set of bananas’ and {banana 1, banana 2, ...}. The first is not fully defined.

Or the ‘set of rationals’ - not defined and is undefinable

it's two ways of defining instances of the same concept. — MindForged

But the two ways produce two different concepts which maths tries to then treat in the same way via fudges like transfinite numbers.

https://en.wikipedia.org/wiki/Set_(mathematics)

So you are allowed to define a set:

- intensionally. By specifying selection criteria
Or
- extensionally. By listing each member.

These are two different definitions of the same core concept ‘set’. Using one label ‘set’ for two distinct concepts is bound to lead to confusion.

Intensional definition also allows an incomplete definition of a set such as ‘the set of all bananas’ - that is only a partial discription so the set is UNDEFINED.

You don't have to list out all the members of a set to define it. Seriously, sets are defined intensionally all the time. — MindForged

But a set is a list of elements, if you don’t list the elements you are missing out the definition of the set.

When we say ‘the set of bananas’ we are not defining a set, just specifying the selection criteria for the set which is a different thing from the actual set.

For example the actual set of bananas has a cardinality so clearly the actual set definition contains more information than the selection criteria.

What is sound about the ‘set of all sets does not exist’? It exists as much ‘as the set of Naturals’ yet it does not exist in set theory.

But anyway, neither of the above are fully defined sets. You have to list all the members to fully define a set.

https://en.wikipedia.org/wiki/Measure_problem_(cosmology)

Another infinity paradox. In this case cosmologists are plugging in Actual Infinity for the size of the universe into probability and getting nonsense like ‘two headed cows are as likely as one headed cows’

Yes so it might be that the being with greatest good has only say 97% good. That does not square with some of the traditional views of god...

Infinity is provably not part of the material world (discussed at length here https://thephilosophyforum.com/discussion/4073/do-you-believe-in-the-actually-infinite).

The non-material, if it exists, is less constrained...

P1 argument does not cut it for me; just because a maximum possible quantity exists it does not follow there is an object with such a quantity.

Now the prime mover, I mostly buy...

- Potentially infinite is the process of continued and potentially endless iteration (IE a limit).
- Actually Infinite is the result of an unbounded number of iterations; IE NOT DEFINED (IE an infinite set)

BTW the universe is a macroscopic object, the question of whether it has starts and ends is a macroscopic question. Quantum Mechanics is a microscopic theory so has no bearing on this particular question.

No I believe the universe has an end: The universe is a material object and material objects have starts and ends.

- The concept of potential infinity is useful as an approximation of the very large and small. Potential Infinity exists in the material world.

- The concept of actually infinite is not useful and does not exist in the material world.

I understand it:

https://en.m.wikipedia.org/wiki/Actual_infinity

I understand that you do not understand what actual infinity is. — Magnus Anderson

Give me one example of the Actually Infinite from the material world.

I understand that maths has tried to build a consistent logical structure around the logical fallacy of the Actually Infinite and has failed. The numerous paradoxes attest to that.

so you make a distinction between something you call "Absolute" infinity and any other sort of infinity. I don't know what that difference is, — SophistiCat

https://en.m.wikipedia.org/wiki/Actual_infinity

Please read the definitions of ‘Actual’ and ‘Potentially’ Infinite are very helpful.

I'm assuming a sphere. Or is a sphere not possible? — tim wood

We were looking for examples of actual infinity in nature
You said a sphere with infinite segments
But it’s not proven that nature is continuous
So your sphere can’t have infinite line segments
For the purposes of this proof

A decimal representation has a fixed number of digits; always will have a fixed number of digits. No need for actual infinity.

I think it’s an early attempt at building a metaphysical model of the universe

You are assuming space is continuous to get smooth sphere on which to plot. IE you are assuming what you want to prove.

You do not have a truely continuous sphere on which to make such a construction.

You also have no geometrical equipment with precision 1/oo to make the construction

The very fabric of space time is stretching. It does not need anywhere to expand into; time and space don’t even exist beyond the boundaries

Cantor's Paradox
‘The set of all sets is its own power set. Therefore, the cardinal number of the set of all sets must be bigger than itself.’

The set of all sets is an ACTUAL INFINITY so not a completly described set. You cannot soundly reason with it. Leads to the paradox.

Not sure this is much better:

- There is an number X such that X > all N
- X+1 > X
- There is no such number

The problem, as you pointed out above, with the preceding argument is that there are two sorts of numbers involved; finate numbers following the normal rules and infinite numbers in an illogical world of their own.

I think the math is frankly nonsense, how can we operate with two types of different numbers one of which is defined only axiomatically, does not exist in reality, obeys different counter intuitive rules and leads to contradictions?

infinite number is a number that is larger than any finite number. An infinite number is not larger than any infinite number — Magnus Anderson

- So I have infinity X and a copy X’.
- I add one to X
- then X > X’ by common sense

A hotel which is completely full, an infinite number of new guests show up and they are all accommodated by the magic of infinity.

Magic is the key word. How did such a concept find its way into maths? I think it’s historical and relates to our original concept of God - God is omnipotent so must be able to do anything, including the Actually Infinite, so they were thinking.

However it happened we are left with pure and applied math containing spiritualism.

But maths has a responsibility to make sure it clearly communicates concepts to its end users.

Actual Infinity need to come with a health warning:

- This is a conceptual concept only
- Applying it to the real world is nonsense
- It is logically inconsistent with the rest of maths and common sense (see Hilberts Hotel)