Apart from God, all other infinities are potential and theoretical, not actual. The ancient Greek philosopher Aristotle (384– 322 BC) proposed that there are two types of infinity, a potential and an actual infinity. In a potential infinity, one can keep adding or subdividing without end, but one never actually reaches infinity. In a sense, a potential infinity is an endless process that at any point along the way is finite. By contrast, in an actual infinity, the infinite is viewed as a completed totality. Aristotle rejected actual infinity, claiming that only potential infinity exists. Georg Cantor, who invented set theory over 100 years ago, did so in part for theological reasons, seeing the theoretical infinite sets he came up with as a reflection of the actual infinity of God. The legendary mathematician Carl Friedrich Gauss denied that anything infinite really exists, “Infinity is merely a way of speaking” and “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.”
Max Erik Tegmark is a Swedish-American cosmologist and professor at the Massachusetts Institute of Technology. He forcefully argues here that actual infinity within the cosmos is impossible and that while a beautiful concept, is undermining modern physics. "... today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞"
Consider that any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever. Infinity in this case serves merely as a limit which we never attain.
Consider the scenario imagined by al-Ghazali (medieval Islamic theologian who was a forceful proponent of the Kalam cosmological argument for the existence of God) of our solar system’s existing from eternity past, with the orbital periods of the planets being so coordinated that for every one orbit which Saturn completes Jupiter completes 2.5 times as many. If they have been orbiting from eternity, which planet has completed the most orbits? The correct mathematical answer is that they have completed precisely the same number of orbits. But this seems absurd. Think about it: the longer Jupiter and Saturn revolve, the greater becomes the disparity between them, so that they progressively approach a limit at which Saturn has fallen infinitely far behind Jupiter. Yet, being now actually infinite, the number of their respective completed orbits is somehow magically identical. Indeed, they will have “attained” infinity from eternity past: the number of completed orbits is always the same. So Jupiter and Saturn have each completed an infinite number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any finite interval of time. This is impossibly insane.
Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity.
What about God?
When we say that God is infinite we do not mean that He is really, really big in some spatial sense. God has no spatial dimensions; He is spirit (John 4:24). God exists outside space and time – a quality theologians refer to as the “Transcendence of God”. He created space and time (Gen 1:1; John 1:3).
God is above and outside his creation. To be transcendent means to exist beyond and outside the ordinary range of human experience or understanding. God is above, other than, and distinct from all he has made - he transcends it all. Paul says that there is "one God and Father of all, who is over all and through all and in all (Ephesians 4:6). Scripture says elsewhere, "For you, O LORD, are the Most High over all the earth; you are exalted far above all gods (Psalms 97:9; cf. 108:5).
Scripture repeatedly affirms the uniqueness of God. There is no one or anything else like Him. As the only actual infinity, He transcends creation. Thus, unlike His creation, God is not limited by either space or time. He is infinite with respect to space – “omnipresent”; God is simultaneously present everywhere in the Cosmos with all of His fullness. He is infinite with respect to time – “eternal” with no beginning or end.
He is infinite with respect to knowledge – “omniscient” meaning He knows everything that can be known. He never learns anything, adding to His knowledge, not does He ever forget anything. He is infinite with respect to power – “omnipotent” meaning he has all power.
Can God literally do anything? No – i.e., He cannot sin. He can do anything that He wills – in other words He is limited by His own nature. God's power is self-limiting unlike our power which is ultimately limited by external force. To say that God is sovereign is to declare that God is God with the power to be able to do literally anything He wills. That kind of power belongs to God alone (Ex 4:21; Jdg. 14:1–4; 1 Sam. 2:6–8; Ps. 22:28; 24:1; 50:10–12; 139:16; Prov. 21:1; Dan. 4:34–35; Isa 45:7; John 6:44; Acts 4:27-28; Rom 8:29; 9:18; 13:1-2; Eph. 1:4-5, 11).
In fact, God has infinite perfection in all His attributes – love, mercy, justice, wrath, etc.
Because He exists outside space and time, we could not know God had He not chosen to reveal Himself to us. The supreme example of God’s immanence of course is the life, death and resurrection of Jesus Christ, God-incarnate (gospels of Matthew, Mark, Luke & John). His revelation is such that we (as finite beings) can have a meaningful and fulfilling relationship with Him despite His infiniteness.