F there is more than one conclusion and those conclusions are different then we have a contradiction.
No. In maths there are any number of scenarios - the most obvious and basic being solutions to quadratic equations - where there are multiple, equally correct solutions. That's because sometimes, what's being described by the maths, doesn't have a single, unique solution.
If you are defining contradiction differently then you need to explain it.
Differently to you? You're the one making the claim, all I need to do is point out the error in YOUR definition - you are presuming that you can't have multiple valid conclusions from any given situation, that the presence of mathematical ranges or discrete values somehow invalidates the work because you haven't got 'an' answer.
If we look at the citation I gave we see in his working out that he does indeed come up with more than one conclusion when performing the maths of infinity.
Actually, as is spelt out in the introduction, the failure comes from "Some philosophers use Hilbert’s Hotel to argue that actual infinities cannot exist in the physical world because basic arithmetic operations involving infinities lead to “absurdities.” Infinity is not a number, it's a concept; you can do maths with it, but amongst the things you can't do with it is arithmetic.
While that may be logical, if one is arguing that there could be real infinities one has to demonstrate these multiple conclusions occurring physically. And that’s all I am saying.
I write an equation to map the contours of a mountain range - I then solve that equation for all points moving out from, say, Everest's peak, where the gradient turns positive. There are going to be hundreds, possibly thousands or millions depending on the scale to which I'm working and the precision of the equation. All correct, all within the one equation.
However, some “absurdities” arise when groups start leaving. If a finite group of 5 checks out, the hotel still has an infinite number of rooms filled. But consider what happens when two different infinite groups leave the hotel. Having the infinite group in all rooms greater than 5 check out leaves only 5 rooms filled. Alternatively, when the infinite group of all even numbers checks out, the hotel has an infinite number of odd rooms filled. In equation form (paralleling the addition equations above), this gives:
a – b = c Just like addition above
∞ – a = ∞ OK so far
∞ – ∞ = 5 First infinite group leaving
∞ – ∞ = ∞ Second infinite group leaving
Actually, no. Depending on the ranges and the context:
∞ – anything is undefined - you can't do arithmetic with infinity, it's not a number. In mathematical terms, it's like saying 15 - fish.
The last two equations are contradictory! You can’t subtract one value from another value and get two different results.
And here is the crux of the misunderstanding: seven is a value. 'x' in an equation is a value (not currently defined). ∞, though, is not a value. He goes on to explain that, which you even quoted, but then continued with the misunderstanding anyway.
No what I am asking is why physics or nature and not the absence of it.
What makes you think there's a 'why'? You're such a fan of 'necessary entities' - why can't 'laws of nature' be the necessary entity?
If you are claiming that question is incoherent you have to say that physics and nature have to be.
No, just that they are. The probability of an event that's already happened is 1. If there was never a point where physics didn't happen, that probability has always been 1.
O.
