The nominalists gleened from Scotus what would fit their stance, and the realists gleened from Scotus what would fit their stance. They have also done this with Peirce's work. The nominalists and realists were and are both misguided. — Mapping the Medium

With all due respect, this sweeping generalization is rather misleading. Peirce described his own view as "extreme scholastic realism" (CP 8.208; c. 1905), calling himself "an Aristotelian of the scholastic wing, approaching Scotism, but going much farther in the direction of scholastic realism" (CP 5.77n; 1903) and "a scholastic realist of a somewhat extreme stripe" (CP 5.470; 1907). It is not much of a stretch to say that his entire philosophy is the result of a lifelong crusade against nominalism, and there was no more disparaging statement from his pen than the charge that someone else was a nominalist.

I have sometimes wondered about the length of 'now' - it seems it cannot be zero length else 'now' would be nothing (nothing length zero has existence). This type of thinking leads to consideration that time maybe discrete and eternal, over which the is much controversy. — Devans99

On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot have zero duration requires time to be continuous, such that "now" has a duration that is infinitesimal--not zero, yet less than any assignable or measurable value.

The same rules apply for finite and infinite sets. — Devans99

As long as you continue to insist on this, there is nothing more for us to discuss.

A googol-sized set could have logical existence if our universe was bigger. — Devans99

By that reasoning, an infinite set could have logical existence if our universe was infinite. But logical possibility is not at all dependent on actuality, or even metaphysical possibility. So infinite sets do exist, in the strictly mathematical sense of existence.

A greater than any number sized set does not even have logical existence (leads to absurdities so it cannot be logically sound). — Devans99

It only leads to absurdities if one insists on attempting to apply the same rules to infinite sets as to finite sets. Mathematicians have long recognized this, which is why there are different rules for infinite sets.

To be clear, since this is a common misunderstanding: The Planck scale is the point at which our theories of physics break down and may no longer be applied. It does NOT mean the world itself is quantized. Below the Planck length we simply do not know and can't even speculate, because our physics no longer works. — fishfry

Thanks for this. I would add that the same is true of the Planck time, since it is defined as the duration required for light to travel the Planck length in a vacuum.

Please provide citations or (better) quotes from Peirce's writings to substantiate your assertions about his views, as I did for mine. I remain unconvinced that you have carefully studied them to gain a thorough understanding of his expansive mathematical, phenomenological, semeiotic, and metaphysical thought. Distinct points/instants are indeed arbitrary and artificial creations of thought, but indefinite infinitesimals/moments are real, with length/duration less than any assignable value and no discernible boundaries. We can only mark points/instants to divide lines/time into finite segments/lapses.

Let me rephrase that then, Peirce replaces the "point" with the "infinitesimal", as the point might be designated as unreal, and incapable of producing a continuity. — Metaphysician Undercover

Peirce does not "replace" points with infinitesimals; they are two different concepts, and there is still a role for points--not as the parts of a line, but as the discrete boundaries between its continuous parts. He helpfully clarifies this in one manuscript (R 144, c. 1900) by referring to points (or instants) as limits and the line segments (or lapses of time) between them as portions. In later writings he reverts to "parts" for the latter, but suggests "connections" for the former.

The duration of time cannot consist of "instants", or points, which have zero duration, but it may consist of "infinitesimals", which I might have carelessly referred to as points with extension. — Metaphysician Undercover

The duration of time does not consist of infinitesimal moments, either. As with anything truly continuous, the whole is ontologically prior to any of its parts.

... I conceive that a Continuum has, IN ITSELF, no definite parts, although to endow it with definite parts of no matter what multitude, and even parts of lesser dimensionality down to absolute simplicity, it is only necessary that these should be marked off, and although even the operation of thought suffices to impart an approach to definiteness of parts of any multitude we please.*
*This indubitably proves that the possession of parts by a continuum is not a real character of it. For the real is that whose being one way or another does not depend upon how individual persons may imagine it to be. — Peirce, RS 30, 1906

The parts are indefinite (infinitesimals/moments) unless and until we arbitrarily mark them off (with points/instants).

Each infinitesimal requires a point of division, a boundary, to separate it from another infinitesimal. — Metaphysician Undercover

No, the very nature of infinitesimals/moments is that they are not distinct from one another at all.

Another plain deliverance of the percipuum is that moment melts into moment. That is to say, moments may be so related as not to be entirely separate and yet not be the same. Obviously, this would be so according to our interpretation. But if time consists of instants, each instant is exactly what it is and is absolutely not any other. In particular, any two real quantities differ by a finite amount. — Peirce, CP 7.656, 1903

We can only introduce points/instants as the boundaries between adjacent segments/lapses that have finite length/duration.

If infinitesimals are real, this provides the logical foundation for the reality of continuity. But the only thing which supports the reality of the infinitesimals is the "need" to support the continuity. — Metaphysician Undercover

If we have good reason from our phenomenal experience to posit that continuity is real, and the hypothesis of infinitesimals "provides the logical foundation for the reality of continuity," then we have good reason to conclude that infinitesimals are likewise real.

Now a fact that goes to show that time is continuous is that our consciousness seems to flow in time. If we suppose that we are immediately sensible of time, the origin of the idea is explained; but if not, then we must cast about for some other way of accounting for our having the idea. Now there are great difficulties in the way of supposing that we are immediately conscious of time, and therefore of the past and future, unless we suppose it to possess the third property of continuity [infinitesimals], so that we can be immediately conscious of all that is within an infinitesimal interval from any instant of which we are immediately conscious, without its thereby following that we are immediately conscious of all instants. — Peirce, R 257, c. 1894

What is the argument for denying the reality of infinitesimals?

See, the problem here is that the "two instantaneous states" are not real. — Metaphysician Undercover

That is not a problem at all, it is precisely Peirce's view.

... it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion ... — Peirce, R 295, 1906

Instantaneous states are creations of thought for describing real events in time. We arbitrarily mark them at finite intervals, but the reality is continuous motion/change.

But if divisions in time are created artificially by positing such points, then there is no principle to deny dividing time infinitely. So the infinitesimals are posited solely for the purpose of denying infinite division, without any real substance. — Metaphysician Undercover

Nonsense, Peirce consistently affirms that time is (potentially, not actually) infinitely divisible, and that this is always necessary (but insufficient) for true continuity. In fact, he asserts repeatedly that instants of any multitude, or even exceeding all multitude, may be inserted within any lapse of time--even an infinitesimal moment.

In order to have real existence, the infinitesimals require real boundaries. So if the infinitesimals are real, then the continuity is not, due to the existence of the boundaries. — Metaphysician Undercover

This is a confusion of reality and existence. Indefinite infinitesimals/moments are real, but do not exist; distinct points/instants exist, but only by virtue of being marked off by an act of someone's will.

Again I confess I don't know what you mean by real versus actual. Can you give an example? — fishfry

I am largely employing the terminology and philosophy of Charles Sanders Peirce in all this, so I will offer a couple of his examples. If I were to hold a stone in my hand and then release it, then it would fall to the floor. This subjunctive conditional proposition represents a real law--one that is true regardless of whether I ever actually carry out the experiment that it describes. Similarly, any diamond really possesses the character of hardness, regardless of whether anyone ever actually scratches it with corundum to demonstrate that it does.

I do not know enough about public key cryptography to hazard a guess at how the distinction between reality and actuality applies to it. In general, I believe that the practical effectiveness of mathematics stems from its hypothetical nature; the key is that the idealized model must adequately capture the significant aspects of the actual situation. I am a structural engineer, so I routinely use a computer to simulate the effects of gravity, wind, earthquake, etc. on a building in order to design it such that it can be expected to remain standing once actually constructed.

It says there exists such a set — Devans99

Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a member of the set of natural numbers; but according to physics, the total quantity of actual particles in the entire universe is only about 10^80.

What we are all taught at school - the Dedekind-Cantor continuum - a line is an actual infinite set of points - is an actual infinity. — Devans99

Okay, but you and I agree that this now-standard mathematical definition of a continuum is philosophically faulty; in my case, because I hold that a line is not composed of an actually infinite set of points. Instead, the continuous whole is ontologically prior to any discrete parts, which only become actual when we arbitrarily mark off a finite quantity of them for some purpose.

Something that has potential but not actual existence is not well defined. — Devans99

Please provide an authoritative source for this claim, or just acknowledge that you made it up. Here is what "well-defined" means in this specific context.

• "A set is well-defined if there is no ambiguity as to whether or not an object belongs to it, i.e., a set is defined so that we can always tell what is and what is not a member of the set." (source)
• "In mathematics, a well-defined set clearly indicates what is a member of the set and what is not." (source)

By virtue of the procedure by which we could logically go about constructing the set of all natural numbers, it clearly qualifies as well-defined in the relevant sense. Even though there is a potential infinity of its members, such that we could never actually assemble the complete set, we can always easily determine whether any proposed candidate is or is not one of those members. 5 is, but 5.1 is not. 750,943,179,981,061 is, but a banana is not.

So, can you provide an example of something whose membership in the set of all natural numbers is ambiguous? That would be the only way to demonstrate that it is not well-defined.

Some parts of math are obviously true, or actual. — fishfry

In my view, nothing within mathematics is actual--again, it is the science that reasons necessarily about strictly hypothetical states of affairs--and truth has to do with what is real, not just what is actual. Specifically, a proposition is true if and only if its subjects denote real objects and its predicate signifies a real relation among those objects. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it, while the actual is that which acts on and reacts with other things.

But there are truths that aren't physical. "5 is prime" is one of them. — fishfry

We agree on this--the number 5 and the character of being prime are real, even though they are not actual.

But "5 is prime" IS true in the actual world. — fishfry

How so? Given my definitions above, perhaps what you mean is that "5 is prime" is true in the real world; in which case, again, we agree on this.

Cantor did claim actual infinity exists: — Devans99

Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief that there is an "actually infinite number of created individuals" does not somehow falsify all of his mathematical ideas about infinity.

How exactly can the set of naturals be potentially infinite? — Devans99

We can prescribe how you would logically go about constructing the set of all natural numbers, but we cannot actually carry that process out to its completion.

It is defined as an actual infinity (all sets are actual). — Devans99

Please provide an authoritative reference for the claim that "all sets are actual." Remember, mathematical existence does not entail metaphysical actuality.

The set of natural is defined in maths as an actual infinity
'The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers' — Devans99

No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an extreme platonist, claims that there is an actual set containing all the natural numbers.

Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist

Do you think that "5 is prime" is true? — fishfry

Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. Do you think that either of these terms denotes something actual?

You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual impossibility does not entail logical impossibility. Mathematical existence is not metaphysical actuality. The infinity of the natural numbers is potential, not actual. Continuity of space does not require an actual infinity of distinct positions.

I believe that the naturals and reals are purely mental constructs. They exist in our minds only (where the impossible is possible). They have the same status as talking trees and levitation - illogical/impossible things can exist in our minds but they cannot exist in reality. — Devans99

All of these are logically possible, just not metaphysically possible.

An actual infinity of naturals (IE a set with a greater than any number of elements) is impossible. — Devans99

No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity.

It is not logically possible to complete a task that has no end. — Devans99

Incorrect--it is logically possible, just not metaphysically possible.

Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'. — Devans99

This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual.

Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible. — Devans99

This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible.

In summary, mathematics is the science that draws necessary conclusions about strictly hypothetical states of affairs. That includes its application to infinity--never actual infinity, always potential infinity.

If space is continuous then my hand moves through an actually infinite number of intermediate positions. — Devans99

False. Again, if space is a continuous whole, then it is not composed of individual and distinct positions.

Paradoxes indicate we have a wrong assumption somewhere, in this case, the assumption that it is possible to complete an actually infinite number of steps in a finite time is suspect. So I doubt that true continuity is possible. — Devans99

The wrong assumption in this case is that the true continuity of space would require your hand to complete an actually infinite number of steps by passing through an actually infinite number of intermediate positions. As I have explained repeatedly now, the only individual positions that exist are whatever finite quantity of them we explicitly mark. If you still want to insist that real space is discrete, then make your case, but please stop pretending that this particular objection to its continuity is valid.

If my hand moves from position 0 to 1, it is guaranteed to pass through positions 0.5, 0.25, 0.125, etc... Or are you saying it somehow skips over intermediate positions? That would be discrete movement. — Devans99

No, the only actual intermediate positions are the ones that we individually mark. There is a potential infinity of such positions, but we can only mark (and thereby actualize) a finite quantity of them. Again, continuous motion is the reality, while discrete positions are our invention.

But movement is something that actually happened in the past - my hand in the past moved through all possible positions - so that must be an actual infinity of positions. — Devans99

Given whatever path your hand actually followed in moving between its initial and final positions, it indeed moved through all possible intermediate positions along that particular path--again, a potential infinity, not an actual infinity. In order to describe that path, we would need to mark various intermediate positions and assign coordinates to them relative to the initial and final positions, taking the distance between them as our arbitrary unit of length. The more positions we mark and measure, the more accurate the resulting description of the motion--but it will never be perfectly accurate, since we will never be able to mark and measure all the possible positions; merely a finite quantity of them, which are the only actual positions.

But the action of movement does mark positions 0 and 1 and all positions in-between. — Devans99

No, it only marks positions 0 and 1; marking any individual intermediate positions would require their explicit designation, and we can only ever do that for a finite quantity of them.

We know that our hand actually passed through all those positions in the past, so if space is a continuum then motion actualises an infinity of positions. — Devans99

No, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion.

As pointed out in the OP, actual infinity is absurd, so therefore space is not a continuum. — Devans99

No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions.

The problem with Peirce's metaphysics is that he allows that pure, absolute continuity, which can only be expressed by us human beings through the terms of infinity, to be polluted by the concept of "infinitesimal". — Metaphysician Undercover

How much of Peirce's metaphysics (and mathematics, and phenomenology, and logic/semeiotic) have you actually studied carefully? What fundamental distinction are you positing here between the concepts of "infinity" and "infinitesimal"?

A succession of infinitesimal points does not provide the necessary conditions to fulfil the criteria of "continuity". — Metaphysician Undercover

Peirce would agree with this, although "infinitesimal point" is a contradiction in terms. There are infinitesimals, and there are points; they are two very different concepts, since infinitesimals have extension (though smaller than any assignable/measurable value), while points do not. His parallel terms when discussing time are moments, which have duration (though shorter than any assignable/measurable value), and instants, which do not.

Positing a degree of difference as existing between the infinitesimal points, no matter how large or small that degree of difference is, necessitates the conclusion that there is something "change", which occurs between such points, rendering the supposed continuity as non-continuous. — Metaphysician Undercover

Peirce would agree with this, as well. Infinitesimals (and moments) are indefinite, and thus cannot be individually distinguished; we can only discern differences once we have marked off specific points (or instants). In fact, one of Peirce's own definitions of a moment is "a time in which no change which can in any way be made sensible can take place." A finite lapse of time between two marked instants is required for any difference to become discernible. In his own words, "between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false [Cantorian] continuity which the calculus recognizes but in a much stricter sense."

Because Peirce proposes a polluted, and impure form of continuity, rather than starting with a pure and true continuity as his first principle, his approach to agapasm is demonstrably a materialist approach. — Metaphysician Undercover

Peirce would vehemently deny both charges here--he does start with a pure and true continuity as his first principle, or at least consistently strives to do so; and he explicitly rejects materialism, calling it "quite as repugnant to scientific logic as to common sense," instead affirming objective idealism as "the one intelligible theory of the universe." It treats "the physical law as derived and special, the psychical law alone as primordial," such that "matter is effete mind, inveterate habits becoming physical laws." Accordingly, Peirce's cosmology understands the very constitution of being as true continuity underlying indefinite possibilities, some of which are actualized by the ongoing process of determination.

If my hand passes from position 0 to 1, then it passes through positions 1/2, 1/4, 1/8, 1/16, ... 1/∞. — Devans99

This is the same mistake that Zeno made. Positions 0 and 1 do not exist unless and until we arbitrarily mark them as such, and the same is true of any and all intermediate positions between them. We cannot proceed to mark an actual infinity of those; we can only hypothesize that there is a potential infinity of such positions in accordance with our arbitrary system of measurement--for example, the real numbers. Moreover, the physical distance from 0 to 1 does not matter--whether it is 1 kilometer, 1 meter, or 1 millimeter, the real numbers assign the same multitude of intermediate positions between them; likewise if we change our designation of the second position from 1 to 2 or any other value. All this demonstrates that real space is continuous; we only model it as discrete for practical purposes.

Assuming for the sake of argument space is a continuum (which I do not believe), when I move my hand from left to right, it would pass through an actual infinity of distinct positions. — Devans99

No, it would not. If space is truly continuous, then it is not composed of distinct positions. We arbitrarily impose distinct positions on space for various purposes, including measurement. They are entia rationis, creations of the mind, not constituents of reality itself.

Or some hold that space is continuous, implying an actually infinite collection of distinct spacial positions in a unit of space. — Devans99

Only those who (perhaps naively) embrace Cantor's mathematical model of a continuum as isomorphic to the real numbers would affirm this metaphysical implication, thus claiming that space is somehow composed of dimensionless points. Such a collection of distinct objects is a bottom-up construction in which the parts are ontologically prior to the whole. By contrast, I would argue that true continuity is a top-down construction in which the whole is ontologically prior to the parts. Positions are only actual if and when they are marked for a purpose, such as measurement; otherwise, they are strictly potential.

I don't think there is a widespread misconception. I think there's a widespread lack of interest in the question; and among those who are interested, some degree of agreement that the real numbers don't express everything we think must be true about a continuum. — fishfry

Fair enough, thanks.

There's no such thing as real space and times. — Metaphysician Undercover

If you adamantly deny the reality of space and time, then there is nothing more for us to discuss on that front.

You can't say that the mathematical definition is wrong, because it's a mathematical term. — Metaphysician Undercover

The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not.

What we call "space" and what we call "time", are abstract ideas created to account for what we observe. — Metaphysician Undercover

This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves.

"Continuum" as implied by common usage means a collection of contiguous, but separate individual units. — Metaphysician Undercover

That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. As George Box famously put it, "All models are wrong, but some are useful."

"Continuum" and "continuity" have very different meanings. — Metaphysician Undercover

Only in the sense that continuity is a property, while a continuum is anything possessing that property.

Peirce's rather poetic analogy between semeiosis and music in "How to Make Our Ideas Clear" (1878) seems relevant here: "In a piece of music there are the separate notes, and there is the air ... Thought is a thread of melody running through the succession of our sensations ... [Belief] is the demi-cadence which closes a musical phrase in the symphony of our intellectual life."

I think modern science has demonstrated that there is a smallest unit of space. — Metaphysician Undercover

This is a common misconception. What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself.

As for the thread title and OP, Cantor's analytical definition provides an adequate model for most mathematical and practical purposes. However, it is a bottom-up construction that wrongly attempts to assemble a continuum from distinct parts corresponding to the real numbers. In other words, it presupposes that any line, surface, or solid of any length, area, or volume is composed of all such parts--namely, points--which is why the Banach-Tarski theorem follows from it.

By contrast, a true continuum is a top-down conception in which the whole is ontologically prior to its parts, all of which have parts of the same kind and the same mode of immediate connection to each other. Those parts are indefinite unless and until we arbitrarily mark them off for a particular purpose, such as measurement, by inserting limits of lower dimensionality, which can hypothetically be of any multitude or even exceed all multitude. A line is composed of lines that are contiguous at points, a surface is composed of surfaces that are contiguous at lines, and a solid is composed of solids that are contiguous at surfaces.

There's a proper class of Alephs, indexed by the proper class of ordinal numbers. After ℵ0,ℵ1,ℵ2,… come ℵω,ℵω+1, and onward forever; too many Alephs to ever be captured in a set. — fishfry

Thanks for the correction.

Please define your terms and assumptions. What is "time"? What is a "moment"? Why do you think that "infinite time between any two moments" is impossible?

I was thinking of the generalized continuum hypothesis--the idea that for any infinite set of a given cardinal (aleph_n), its power set is always of the next higher cardinal (2^aleph_n = aleph_n+1), with no cardinals in between and no largest such cardinal. Those cardinals are obviously in one-to-one correspondence with the natural numbers; i.e., countably infinite.

In this case there is no underlying reality. There is no "setness" that we are trying to formalize. Rather, sets are whatever satisfies the rules we're writing down. Before the rules are written down, there are no sets! — fishfry

Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things.

What he [Cantor] did do was develop ideas of transfinite cardinals and ordinals along with an arithmetic of transfinite cardinal and ordinal numbers. That is, of different sized infinities, of which there are apparently a whole lot. — tim wood

Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable.

The universe is orders of magnitude more complex in its order, therefore requiring a designer. — TheMadFool

No, mere complexity is also insufficient; "intelligent design" theory requires specified complexity and/or irreducible complexity to count as evidence of design. Again, the plausibility of such an approach depends on one's opinion of the underlying assumptions.

The relevant feature between the universe and a watch is order - a specific arrangement of parts following a set of principles/laws. — TheMadFool

"Intelligent design" theory acknowledges that mere order is insufficient, instead requiring "specified complexity" to count as evidence of design. Its proponents cite well-established scientific fields, such as forensics and archaeology, that have particular methods and criteria for distinguishing intentional agency from natural processes.

However, as with any hypothesis, plausibility depends on acceptance of the underlying assumptions, and an objection such as that of carries some weight. Of course, we also have no experience with universes popping into existence or entirely new kinds of animals evolving, so it cuts both ways to a certain extent.

If Craig was to declare "Yahweh does not exist", then how would that be any different from atheism? — jorndoe

Charles Sanders Peirce denied the existence of God, but argued for the reality of God. Something exists if it reacts with the other like things in the environment; something is real if it has characters regardless of what any individual mind or finite group of minds thinks about it. Existence is spatio-temporal, but reality need not be. Everything that exists is real, but there are realities that do not exist (in this sense).

If God is not the creator of logic, then logic would be primary to God. This is clearly problematic if God is to exist eternally. Therefore, God must be the creator of logic. — Teaisnice

Or, logic is simply consistent with the eternal and immutable nature of God, rather than something primary to God or created by God. This is basically a variation of the Euthyphro Dilemma, and it is exposed as a false dichotomy in the same way.

Either God can do all things or he is limited to do only logically possible things. — Teaisnice

It is not a limitation of God to be capable of doing all logically possible things, since it is God's eternal and immutable nature that determines what is logically possible.

This explanation is just the sketch of an argument by contradiction for the Gödel-Carnap's diagonal lemma — alcontali

Thanks for this. Just curious, is there a constructive argument for the lemma, such that it also holds for intuitionistic logic where proof by contradiction is not allowed?

How come religious people like God when these same qualities are disliked when in their comrades at an infinitely smaller scale? — TheMadFool

What we tend to dislike about certain people is not their actual (finite and imperfect) love, power, or knowledge, but their ways of expressing their love, their lust for and/or abuse of their power, and their arrogance about their knowledge.