1. Originally Posted by PyramidHead  Originally Posted by beero1000 As far as I can tell, there's no way we could measure a property as infinite. Any gauge made by humans will have a finite size, and thus should only be able to encode a finite amount of information.

But that doesn't necessarily preclude infinity as a scientific possibility.

Let's suppose there is a location in the universe where some property actually is infinite. Nearby, the measurement is high and as you get closer to the singularity, it gets higher and higher. Until your gauge overloads and breaks.

We'll never be able to actually measure 'infinity' on a tool, but that doesn't preclude us from concluding that that an infinite measurement exists. It's a straightforward, falsifiable, scientific statement. In order to falsify it, an experimenter just needs make a tool that can withstand and measure the property at a finite value. If it overloads every tool we can make, is it reasonable to hypothesize that the value is infinite?
Not necessarily, because there is always the possibility that our measurement tools are thousands of orders of magnitude too crude. Depending on the phenomenon, of course. The fact that we don't have a definite measurement of something is only evidence that it's infinite inasmuch as having a definite measurement would prove that it's not infinite. But that's like saying the Pope's claim of infallibility is evidence that he is infallible, since if he said he wasn't infallible that would prove he wasn't. In other words, being too high to measure is merely consistent with being infinite, but it's also consistent with a great many other possibilities that can't really be discarded in favor of infinity in any principled way.
There's always the possibility that I can fly. The fact that I don't have definite proof that I can't doesn't prove that I can.

Science isn't about considering every possibility - it's about falsifiably explaining the evidence. For any finite value there are higher finite values, so in this hypothetical, how high must we be able to measure before we can conclude the value is infinite? What if there's theoretical evidence supporting it, say like a singularity in a well-supported theory?  Reply With Quote

2. A set is basically a data structure equipped with an elementOf algorithm that returns true or false depending on whether or not something belongs to it.

The axiom of infinity asserts that there is a set whose elementOf algorithm returns true when given the empty set as input, and that returns true on x cup {x} whenever it returns true on {x}. The other axioms of set theory assert there is a set whose elementOf always returns false (i.e., empty set exists), give various ways to create sets with elementOf algorithms based on existing sets' elementOf algorithms, and prevent (via the foundation axiom) having an elementOf which always returns true.

Typically we choose our physical models to try to minimize the size of both the model and the number of bits needed to specify how it disagrees with observation. Our physical models are specified in terms of mathematics which can in turn be formalized in set theory.

Specifying the elementOf algorithms sufficiently for the sets involved to do calculations can be done with a fixed finite number of bits. If the choice of elementOf algorithms for the best physical theory is more succinctly described if some of its sets' elementOf algorithms satisfy the axiom of infinity, then to me that would argue that reality is best described using infinite sets.  Reply With Quote

3. The concept of finiteness seems to inescapably imply infinity; ending implies endlessness. But that isn't the same as measuring infinity.

Semantically, the concepts of measurement and infinity seem incommensurable; the definite cannot be infinite, nor vice versa.  Reply With Quote

4. Originally Posted by Jobar The concept of finiteness seems to inescapably imply infinity; ending implies endlessness. But that isn't the same as measuring infinity.

Semantically, the concepts of measurement and infinity seem incommensurable; the definite cannot be infinite, nor vice versa.
Finiteness does not imply infinity.

It implies finiteness.

Infinity is an imaginary concept we sometimes apply to finiteness.

We apply this imaginary concept of endlessness to finite things.

But there is no reality except the finiteness.  Reply With Quote

5. Conceptually infinite and quantifiable are mutually exclusive.

On a continuous probability distribution it is not possible to determine an exact probability.  Reply With Quote

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