But three things at least remain controversial.

Draw a circle in ink. You now have two things, substance/ink and form/circle. If there are two things you can separate them - that is, if the circle, as an abstraction, can exist without substance. But how can you do this? Can you lift the abstraction away from the ink so you can hold up an abstract circle? What would that look like? If there is no substance the universe is an abstraction. But abstractions cannot exist without, at least, a mind to keep them in being.

Space? Space is a positive existence, it is not nothingness.

But learn the differences between belief and reality, what is true and what is supposed. — tim wood

It is not a supposition that form/contingency, must have substance, otherwise the universe is an abstraction.

Is this a definition, or an observation? — Banno

It is a deduction. There can be no properties/contingent things, without substance. The eternal substance that is, is the substance of all contingent things. You cannot have a property/form without substance.

So you hold that God exists, yes? Can you say anything substantive about
that existence? — tim wood

God is substance. One of the most difficult questions in philosophy is why there is something rather than nothing. We don't know why or how but we know there is something. That substance that is, is eternal (otherwise it came from nothing, impossible). This substance is identical with existence. It is existence. Every contingent thing that exists exists because it is in existence/God. Creation is more than existence, it is being; life, consciousness...

As to intelligently designed, as a home? For whom? For creation? Of what?

Matter is contingent so it is made. It is an image of energy. When creation fell away from God it descended into 'veils of matter' (Origen)

Intelligently designed how or for what? — tim wood

By God and as a home for creation.

That - or something equivalent - would be sufficient empirical evidence. Of course it would be most helpful if the giant voice would also tell us the proper religion: — EricH

God is not going to force faith on anyone. To do that would be to destroy free will. There is no need for God to shout, He will be heard by those who listen.

Indeed they have. My point a while ago. But theirs is based in faith, not existence, not knowledge. — tim wood

There are many arguments based on a reasonable assessment of the situation we find ourselves in. Eg the cosmological argument, the fine tuning argument, etc. The argument that the universe seems intelligently designed because it is intelligently designed is not easy to refute - like the man said 'Sometimes a cigar is a cigar'. Sometimes things really are what they seem to be. Positing multiverses and what not is not an answer. That is, for the most part, pulling rabbits out of hats to make the question go away.

As for claims to personal awareness of God. These are brushed aside with baseless accusations of 'Delusion'. That is not an argument.

and that anything goes- young earth creationism, flat earthism, moon landing denial, anti-vaxxism, and so on. — Enai De A Lukal

You cannot seriously compare theism to flat earthism. Some of the best minds in history have presented very reasonably arguments for theism. It is a reasonable viewpoint.

E.g. modern inflationary cosmology posits a cause for the big bang as we know it, but that cause has a cause, and so on, possibly infinitely back. — Pfhorrest

It is not temporal causal relationships that matter. The question 'What came before the beginning of time?' is almost trivial. Physical Time, like any physical object, is a property of the material universe and ultimately a property of energy. The real question here is 'What are the necessary conditions for contingent things to exist?'

- What does the second turtle rest on? — Olivier5

Something I wrote some time ago...

One of the most intractable questions in philosophy is Why is there something rather than nothing? Very little progress has been made in answering this question. But we know there is something; 'I think therefore I am'. So, at least Descartes exists, or at the very least, 'there is thought' (Bertrand Russell). At any rate, we can begin with the assumption that there is something rather than nothing.
The something that is, is existence. Existence is not a verb, it is a noun. It is the substance that is and always has been. Existence is God and is not contingent upon any previous state. Existence is not a property of anything, rather, existence has properties. To show that existence is not a property assume X has the property 'existence'. In this respect we consider X and existence to be distinct entities (otherwise X is equivalent to existence and there is nothing to prove). We now ask the question; Does X exist (as a distinct entity)? There are two answers;

1. X exists.
If this is the case existence, as a property of X, is superfluous since X exists anyhow. Therefore X is equivalent to existence.
2. X does not exist.

It is incoherent to say a non existent X has properties, let alone the property existence.
This means that if existence is not a property, it is not contingent. That is, not dependent on any previous state. All other realities are properties of existence. The universe is a set of properties of existence. Properties of existence 'inherit' their existence. We can say 'This milk bottle exists'. By this we mean that the milk bottle is a property of existence and the substance of it is existence, because existence is the only substance that is.

In principle we can deconstruct the bottle into glass crystals and we can deconstruct the crystals into molecules, atoms and so on until even the atoms are deconstructed into energy because energy is the substance of matter. It may even be possible to deconstruct energy into a deeper form of energy but this deconstruction cannot go on indefinitely; it cannot be 'turtles all the way down'. We must come to some ultimate substance that supports the properties 'atom', 'molecule', 'crystal' 'bottle'. This substance is that which is from the beginning, existence.
Even though philosophy cannot say why existence is or what it is, there are a number of things we can say about it;

1. Existence is.
2. Existence has vast creative potential because it has emerged into a universe and everything in that universe.
3. It has the potential to become life, because life is found in the universe.

The way in which existence made manifest a physical and mental/spiritual universe is very special. It did not just spew out an amorphous blob of matter. It created the universe in a way that optimized the creative potential of matter. Matter, because of the precise way it is made, is capable of great creative transformations. It can become a planet, a crow, a city, an oak tree. If matter had been just a primitive blob it would not have the immense combinatorial possibilities it has. It is the precise balance within matter and energy that gives the universe its vast creative potential. This is the Fine Tuning Argument.
Arguably, the highest point in the creative evolution of matter is the physical image of life and being; physical creatures. But the physical image is just that, an image. Life and being are of the mind which is non physical.

Existence simply is. Being is concerned with life and consciousness; a milk bottle exists, a creature is alive and conscious. Life is plurality and unity. A single mind in isolation is hardly alive. Life is a discourse between minds, between self and not self. When existence becomes manifest as a myriad of minds it emerges into life and being. Life is the union of God and creation. Through creation existence/God emerge into being and God becomes the living God. Egoism is the severance of unity. The mind's consciousness turns inward and the bonds of life are broken. Egoism is counterfeit being and ultimately, spiritual death.

***

It has been suggested that reality can be nothing more than a circular chain of properties or contingencies supporting each other in a never ending circle with no supporting substance. This is an absurdity as it tries to dispose of any real substance in reality.
A property or contingency is perfectly identified with its supporting substance and cannot be divorced from it. Take for example a bronze sphere. Bronze is the supporting substance, and its sphericity is its property which is contingent upon the existence of the bronze. It is also perfectly identified with the bronze.

Now, suppose you try to separate the bronze substance from its property, its contingent sphericity. You could try melting the bronze in the hope that you would end up with pure, abstract sphericity. But all that would happen is the bronze would turn into a molten puddle and the property, sphericity, would vanish. This shows how the property is perfectly identified with its substance; the property sphericity, in terms of substance, is the bronze.

The idea that there can be a circle of mutually supporting contingencies or properties without substance is suspect.

Let P1, P2,...Pn be a circle of properties. It is asserted that P1 is effectively the supporting 'substance' of P2 and P2 supports P3 and so on until we get to Pn which is the supporting substance of P1, completing the circle. So it goes round in a self supporting circle without any central supporting substance.
But Pn is perfectly identified with Pn-1 in terms of 'substance' (although the substance in this case is not supposed to be actual, it is still assumed to play the part of the supporting substance). Pn-1 is perfectly identified with Pn-2 and so on until we get to-

P2 is perfectly identified with P1.

What this means is that any separation between P1 and P2 is purely conceptual and there is no separation in actuality. Likewise with P2 and P3 etc. In short, P1, P2...Pn all telescope into a single complex property. That is, there can only be primary, not secondary, properties in actuality. That there seems to be secondary properties is only an effect of the abstract conceptual model we make in our minds.

Here is a simple illustration.

Energy is the substance of atoms. P1 = a set of atoms.
Molecules are made of atoms. P2 = a set of molecules.
The cell is made of molecules. P3 = the cell.

Energy is the substance of the whole system because it is the substance of Property 1, a set of atoms. Conceptually we have three levels of properties but working backwards we can see that the cell is a set of molecules. Molecules are a set of atoms. Atoms are energy.
So P1, P2, and P3 can be collapsed into a single complex property, P3, the cell. In real terms P3 is a primary property of energy because the substance of the cell is energy, just as it is the substance of the atom. So, all the in between properties, as such, are just conceptual categories, not actualities. The error in the concept of a circle of a self supporting set of contingencies, without any supporting substance, is that there can be a property without any supporting substance. In other words, the universe is an abstraction, a property, of nothing. But even an abstraction needs a mind to conceive it and keep it in existence.

Even if that was true, which it isn’t, how do you decide where to stop and say “this is the last turtle”? How do you know your God is the last turtle, or that energy or spacetime or the inflation field or something like that isn’t? — Pfhorrest

What matters is that there is an initial substance. Contingent things are properties of that substance. As I said, first steps first. What I'm trying to establish first is that the universe is made/contingent. And it must be contingent upon some ultimate substance.

The question then becomes What is that substance? If it is a 'mind' and is conscious and intelligent then that is close to a definition of the traditional God.

And just because everything comes from something doesn’t mean there is one particular thing from which all other things come; each thing can come from a different thing. — Pfhorrest

That is unlikely because then you would have numerous sources and numerous eternal existences but then why would they be so congruent to each other? All these contingent things are the same in the sense that the are in the same reality and obey the laws of nature. They are made of the same stuff as far as we can tell. Tea cups and stars are made of the same thing, energy. So are mountains and oak trees. On the level of energy the universe is one thing, a field of energy.

But you’ll also say that graphite is contingent, no? — Pfhorrest

Yes, graphite, being a physical substance, is a property or is contingent.

And is that energy contingent or not? — Pfhorrest

Maybe it is. If energy, as we define it, is contingent upon some deeper substance/energy then that deeper substance is the substance of energy. But there may be an even deeper substance and so on for a bit. But only for a bit. The process of deconstruction cannot continue forever. It cannot be 'turtles all the way down'.

A hydrogen atom is a concept or state. As such it is a physical image of energy. All material constructions are images of energy. That is because they are contingent. The Cosmological Argument says that contingent things must have a beginning. Otherwise there are only states without substance and I don't see how that can be.

It's not my argument. And no one claims you cannot know God completely. The argument is that "God" as defined/understood cannot be known at all. — tim wood

Well, if that's the argument it is wrong. Many mystics have known God's Presence.

This is the main point of contention that you seemed to just brush past in the first part of your reply. Why can’t it go on forever? Every contingent thing has a source. Sometimes that source is another contingent thing. Which might in turn be sourced to another contingent thing. Why at some point must it be different? Why not an infinite string of contingent things sourced from other continent things? — Pfhorrest

Contingent things are essentially properties. A property depends on substance to exist. Take a circle. It is a mathematical abstraction. As such you can't hold it up and show it to anyone; it does not exist. To make it exist you need to give it substance; draw it with graphite. Now the graphite is the substance of the concept of a circle and, having substance, the circle is made manifest. Contingent things are essentially properties, they cannot exist without substance and a regression of substanceless properties cannot exist because a property is a condition of some preceding existence. You are saying that reality can be made of a regression of essentially abstract properties without substance.

Besides, a regression of properties, p1, p2, p3, ... is really just one property. For example, you have a substance, energy and a property of energy, matter. This is how it goes:-

Energy -> p1 (atoms)
Energy -> p1 - > p2 (molecules)
Energy -> p1 - > p2 - > p3 (a cell)

But the distinction between atoms, molecules, and a cell is only by way of classification. In real terms there is only one property (p1 U p2 U p3) = P so

Energy -> P (cell)

The cell is just one complex property of energy. As far as contingency and properties are concerned P (cell) is all that is needed.

They reasoned, in brief, that God is unknowable. — tim wood

Faith also teaches that it is possible to know God. So 'unknowable' must be qualified. Faith says that it is possible to know God as a person but that does not mitigate against the argument that God is ultimately unknowable by human beings. In other words, God can be known as a person but not fully knowable by the human mind.

The claim, then, that God exists is for oh, so many reasons an exhibition of ignorance at the very best. — tim wood

Claiming that God exists and claiming that God is (completely) knowable are not the same thing.

What is each contingent thing has as its source some other contingent thing, such that there are no non-contingent things? What then is God? — Pfhorrest

Contingent means there must be a preceding substance. For example, matter cannot exist without the preexistence of energy. It is not possible to have a property (contingent thing) without substance. Properties can only be a property of a substance in the way that matter is (sometimes) a property of energy

If the Big Bang turns out to be the source of all contingent things, and is not a person or in any way at all like a mythical deity, just some impersonal cosmic event, would you call that "God"? I don't think many would. — Pfhorrest

I'm saying that in order to establish an eternal source it is only necessary to argue that that source/substance must exist. Thereafter it is possible to argue that that substance is mind because the universe is mind-like. It is such because it is mathematical. See also The Fine Tuning Argument. Before talking about God it is easier to establish the existence of a necessary substance and work forward from there and ask questions such as:
Is this substance mind?
Is it a person?
Is it possible for this person to relate to humanity?
Can this person be concerned about what his/her creation does or becomes?
etc.

What I'm getting at here is whether you're okay with the notion of a God that is not a person, that doesn't have thoughts or feelings or wills, that can't hear prayers or issue commands, or judge souls after death, etc. It's just some thing that kicked off existence, and nothing more. Is that really enough to count as "God" to you? — Pfhorrest

No. See last answer.

Must it? [be contingent] Explain. — Pfhorrest

Because matter must have a preexisting substance, energy. Matter is not an ultimate substance, it is a condition that energy is sometimes in. The same may be true of what we call energy; it may be contingent on the existence of some deeper energetic reality. But ultimately this process of deconstruction cannot go on forever, there must be a necessary substance to keep contingent realities in existence.

which is why theism is untenable on any rational or empirical basis, and so is only held as a matter of faith — Enai De A Lukal

Theism is not dependent only on faith. It is a reasonable viewpoint.

God has a lot of company - with an exactly equal claim to existence. Flying hippopotami — tim wood

That is only true for non theists. Theists would disagree. So flat-out atheism must assert theists are misguided/deluded and that claim cannot be convincingly defended. Also, things can be argued to exist through pure reason, without general experience of those things. For example, sub atomic particles can be discovered through mathematical physics. Black holes and other astronomical bodies can be deduced to exist by reason alone (black holes were predicted long before they were an empirical reality). The pure atheist cannot reason in the opposite way and declare that God does not exist.

If the Big Bang turns out to be the source of all contingent things, and is not a person or in any way at all like a mythical deity, just some impersonal cosmic event, would you call that "God"? — Pfhorrest

But the big bang must be contingent. You have to go back further, into eternity, the get to the source.

To start with, the definition of God as the source of all contingent things is sufficient for 'belief in God' and sufficient for a simple definition of God.

As for pure atheism, I don't think it can be defended. We are not in a position to say 'God does not exist'. Such a position, I believe, cannot be defended. Ultimately, agnosticism is the only non theist position.

Physics has shown us the universe has many laws or rules by which it operates; gravitational constant, conservation laws, uncertainty principle, thermodynamics etc — Benj96

The so called laws are not separate from nature. They are merely a description of nature. The electron is not obeying a law that is imposed on it from outside. The law is the electron; it is its nature. Nature is consistent - as far as we can tell - so we formulate this consistency in terms of abstract laws. But there is only nature; it is natural for the electron to behave as it does and what we call law is intrinsic to the electron, it is its nature.

In the beginning there was the inflaton field, and it was without form, and void. — Pfhorrest

There is a difference between nothing and no-thing. Nothingness is just that, no space or time or energy or God or creation. No-thing is no formed thing or no contingent thing or thing that can be defined. This is the void which is the source. The void is existence, in the simplest definition of existence: that which is, necessarily.

Existence is not a property of anything. Contingent realities* are properties of existence.

*Things that are made or dependent on a previous, necessary state.

I really recommend you check out the book I mentioned in my first post. It is actually quite good and the first few chapters are quite approachable. — Tommy

I'll definitely check that one out. Thanks.

Russell suggested we consider the collection "the set of all sets which are not subsets of themselves". Note that, by Frege's abstraction principle, this is necessarily a set. Asking that we, in effect, look the other way and consider instead another set, as you've proposed, doesn't prevent us from considering Russell's set. — Tommy

If I understand you correctly, you are saying "the set of all sets which are not subsets of themselves" is necessarily a set. But it turns out that it is a pathological set or not really a set at all. My argument is that it is an entity of some kind and I'm attempting to define what this entity is. It seems that there can be things that are not simple sets but are infinities of sets. Maybe all sets are really subsets of infinite sets...

Ah, so this is not the correct notation...
Note that we cannot redefine X and that this set is quite distinct from itself. — Tommy

Yes, this is the difficulty with notation but it is really just a notational wrinkle. You can rename the set X' and you have it. All that is required is a set without X as a subset but with all the other sets. The same logic can then be applied to X'₂ and so on. It is really a question of getting the notation right but the concept seems to be coherent.

Well, Russell himself use Type Theory and basically Zermelo-Fraenkel Set Theory (ZF) was made basically to avoid Russell's paradox. If you find your ideas resembling theirs, you can be proud of yourself. — ssu

There are immense complexities with infinite sets so I'm very unsure if my logic holds...

Your definition requests that we posit an object which is logically contradictory, and then remove X from it. — Tommy

I've had great difficulty in this thread with notation but the concept I'm trying to define is simple, namely X\X.

Suppose set A = {a, b} the subsets of this set are:

{{0}, {a}, {b}, {a, b}} Now redefine set A as:

A\A. That is, A\{a, b}

Now the subsets of A are:

{{0}, {a}, {b}}

This is what I mean by X\X.

What's the difference to Russell's Type theory? — ssu

I don't know enough to say but as far as I can see this question can be resolved with simple set theory alone - if my ideas are coherent that is. The problem is with the way the paradox is stated: "The set of all sets that are not members of themselves." Why assume it is a set? Because it turns out that 'The set' is not a set at all. But we can intuitively grasp the concept of 'All sets...etc'. It must be something so what is it if it is not a set? Seemingly it is an infinite collection of sets.
This is similar to Russell's types because there is also a hierarchy of types, each containing the ones below it. So why not just frame the whole thing in terms of sets alone?

The problem with this definition is that the set of all sets that do not contain themselves as subsets is shown, by Russell's Paradox, to be logically contradictory. Your definition requests that we posit an object which is logically contradictory, and then remove
X
X from it. This is akin to requesting the reader to take the smallest prime number with exactly three divisors, subtract it from itself, and then insist that the answer is 0. — Tommy

My argument is to define X without X as a subset of itself regardless of whether it can or can't be such. In this way the paradox is avoided by defining a set that contains 'All sets...' but not X. X is then included in X₂ and the paradox is avoided. The same argument is then applied to X₂, X₃... The result is that the 'Set of all...' is really an infinity of sets each containing the other.

You should consider checking out NBG class-set theory which is an alternative formulation of set theory. — Tommy

My idea is that it can be framed in terms of set theory alone without the invention of classes. My definition is that if X as a subset of itself is excluded this leaves X\X' where X' is the subset.

But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1 — fdrake

This still begs the question about the difference between a limit and an actual infinite sum. Your reasoning shows that you don't run out of natural numbers 'until' infinity.

You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever. — InPitzotl

The reason the unit circle is a standard in geometry is because what applies to the unit applies to a circle of radius 10 or 100 or 1000. It is the internal geometric logic of the thing that matters, not the arbitrary measurement. It is like the difference between inches and centimeters. 1 inch = 2.5 cm. This has more to do with convention than anything fundamental.

If you do. It can also be nothing (0). It can also not be anything. — InPitzotl

But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit width.

Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. — InPitzotl

But you still have 0 + 0 + ... = something

No, it's not arbitrary. It's just infinitely non-specific. — InPitzotl

Say 0 + 0 + 0 + ... = 50 units. Simplify-

(0 + 0 + 0 + ...)/50 = 1

But 0/50 = 0. It's just a matter of reducing to the lowest terms and the same logic obtains.

That doesn’t sum to 1, that sums to 1/9. — Pfhorrest

That was a typo, I meant to say 9/10 + 9/100 + ...

If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined. — InPitzotl

Yes, because it can't be defined in terms of calculus but the question remains, what is it?

But you could do the same thing with a segment of length 2, 50, 0, and -7. — InPitzotl

Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit radius can be 1 inch or 1 light year.

Sorry, what is c here and how does that relate to 0.999...? — InPitzotl

It's the same idea 0.999.. = 9(1/10 + 1/100 + ...)

What seems to be happening here is that 1/x = 0 at infinity.

Then you have the absurd(???) conclusion that

1/infinity = 0

1 = 0 x infinity

So 0 + 0 + ... = 1 after an infinity of terms.

There was an Indian mathematician in the Middle Ages who asserted this (I forget his name)

But most mathematicians probably would not accept this.

But there is also a geometric way to "prove" this.

Take the x,y axis and mark off the unit length from 0 to 1.

This unit represents an infinite string of points all lined up in a straight line.

What is the width of each point? Zero. They are dimensionless.

But they add up to 1 unit width. How do da?

zero width + zero width + ... = extension???

Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. Go figure...

When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1. — InPitzotl

Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? If we're talking about an infinite sum the same applies: by that way of looking at it you would not have 0 < 1/2^c < 1

Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1

But the assertion being made is that 1/10 + 1/100+... can be taken to an infinity of terms and summed to 1. If we take a finite number of terms it won't be 1. That's what I mean by an actual or literal sum. You have to go the whole way.

Such as the vicissitudes of these things. Ramanujan summed the natural numbers and got -1/12.

Calculus is a way of reasoning but there are other ways of reasoning that arrive at different results. Calculus is fine as long as we are talking about finite sums tending towards a limit. But when you go to infinity things get sticky. This is one of the reasons calculus was formulated in terms of finite sums going to the limit, so that the paradoxes at infinity would not interfere.

(a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1. — InPitzotl

Ok, but can't this be also said for 0.999...? Adding terms and then saying 'at infinity'. You can't have (b) at any finite number of the terms 9/10, 9/100...but ya gotta say 'at infinity' sometime if you assert that the literal infinite sum is 1.

I think I got it (incidentally, c=k here, right?) — InPitzotl

Yes, you got it. The point I'm making is that 1/k is positive and > 0. Even as you go to infinity 1/x can't be zero. So, you are summing an infinity of positive terms > 0 which is infinity, right? As the number of terms taken increases 1/x decreases but never becomes zero.

1/2+1/4+1/8+...+1/2^1023+1/2^1024 < 1024+1024+...+1024 — InPitzotl

I'm not talking about the 1024th term. I'm saying-

1/2+1/4+1/8+...+ 1/2^c to k terms - whatever the value of k

>1/2^c + 1/2^c + 1/2^c to k terms. There are the same number of terms in each series.

Contemplate this; I have three dozen pies. I divide them amongst the nine of us. — Banno

Personally I think mathematics is not really about numbers. Mathematics is more about harmonies and proportion. Numbers are 'markers' in the symphony of proportion and relation. The real music of mathematics is beyond numbers. Just a thought...

So that it works at term 10 is irrelevant, because the inequality fails at term 1024 and for all terms after it. You can't go from 10 into infinity without going passing 1024. — InPitzotl

I'm not up to speed on binary. I don't think you understand what I'm saying.
Are all of the terms in 1/2, 1/4, 1/8...positive and > 0? Yes.
For all c 1/2^c is positive and > 0.

Let c run to infinity and sum. Now you have an infinite sum of positive quantities > 0 and that's infinite.

There's no point in saying calculus says otherwise because calculus does not deal explicitly in infinite sums. You need the God calculator for that.

There is no infinite sum of equals on the left side. — InPitzotl

1/2 + 1/4 + 1/8+....+1/1024 > 1/1024 + 1/1024 + 1/1024+....+1/1024

LHS has the same number of terms as RHS

Now let the number of terms run to infinity and the sum on RHS is infinite at infinity.

The next inequality would be-

1/2 + 1/4 + 1/8+....+1/2048 > 1/2048 + 1/2048 + 1/2048+....+1/2048

in fact, they're smaller than 1/2^1024. — InPitzotl

It doesn't matter. An infinite sum of equal infinitesimals must be infinite.

After the 1024th term, we're adding numbers on the left much smaller than 1/1024; in fact, they're smaller than 1/2^1024. And on the right, for each of these, we're just adding 1/1024. — InPitzotl

What is $1/2^c$ added to itself infinitely?

$1/2^c$ added $2^c$ times is $1$

Now add another $2^c$ terms$= 1$

1 + 1 + ... = $\infty$