I understand that you are objecting to the second premise of my argument.

“There is not sufficient evidence for the existence of irrational numbers because they have an infinite number of digits. Yet we theorize they exist as they assist us as an ideal which can be used in the real world.”

Your explanation about numbers being ‘just abstract constructs’ is true, just as the concept of God is an abstract construct. This can be seen in consistently in religious texts with the the Tao Ti Ching of the 6th century BC being one of the first examples-

‘Nameless, is the origin of Heaven and Earth’.

As you say, ‘numbers ‘happen to be useful to describe the world’. Numbers exist because we have a need for them to exist. They assist our survival. In the same way, the concept of God has assisted human survival as well.

As for irrational numbers, I have singled them out as being something that has an extremely abstract concept in that they have no provable end, unlike rational numbers.

Thus, my conclusion:

1. To believe in God is irrational because there is not sufficient evidence for the existence of God.

2. There is not sufficient evidence for the existence of irrational numbers because they have an infinite number of digits. Yet we theorize they exist as they assist us as an ideal which can be used in the real world.

3. In the same way, belief in an infinite God assists us as an ideal which can be used in the real world.

C. Therefore, God exists as irrational numbers exist.

]]>As an example, suppose we can write 2 as a simplified fraction, a/b. a and b cannot have common factors greater than 1, so cannot be both even.

a^2/b^2 = 2, so a^2 = 2 b^2

So a^2 is even, thus a is even

So a = 2k for some integer k

So (2k)^2 = 4k^2 = 2 b^2, so b^2 = 2 k^2

So b^2 is even, thus b is even

But this contradicts the earlier requirement that a, b are not both even

Since our assumption leads to a contradiction, it must be false

So the square root of 2 is irrational.

Of course you may argue that we don’t know these numbers actually exist. It is possible to show irrational numbers exist, using the basic axioms of mathematics. However, these are of course assumptions. But these are not assumptions about the natural world, they are principles from which mathematics can be constructed. Numbers do not physically exist, they are just abstract constructs, that happen to be useful to describe the world, both rational and irrational numbers. So it makes no sense to single out irrational numbers in your argument. You claim that is a type of existence, and that by the same token God must exist – but this does not show that God has any physical existence, but is just a creation of people’s minds.

]]>I think the main quote above was- “No rational number is transcendental and all transcendental numbers are irrational.”

This, I still think is a true statement. All irrational numbers are not transcendental but all transcendental numbers are irrational.

I – “Some irrational numbers are also known as transcendental numbers.”

As for an ‘infinite number’. That is a philosophical question, as these are numbers that are yet to be determined in full. It is infinite in that it has no determined end.

]]>A second error, which permeates your entire discussion, is confusing a real number with its decimal representation. The number π is no more “infinite” than 1/3 is. However, both numbers have a decimal representation that does not terminate in an infinite string of 0’s. This does not make 1/3 “infinite” in any way, and the same thing goes for π .

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