The authors Ellis/ Meissner/ Nicolai published their article The physics of infinity in the magazine Nature Physics, vol 14, August 2018.
The article discusses the (allegedly) spatially finite nature of the Universe based on (known and widely debated) axioms by David Hilbert .
Before I apply my criticism against the article here are two preliminary notes:
- In philosophy and logic we know phenomena of reality that can't be mathematically proven to be true – that's even the rule. Things are as they are.
Only in some cases it becomes scientificly interesting when a physics problem is proved unsolvable. No scientist would come up with the idea because an issue is proved unsolvable it is not part of the Universe.
- In science policy it was a US mission after WW2 to exercise power over German science and particularly to control the German Max Planck Institute for Physics. These strong ties are today not so obvious as they were in the post-war era but they are still there. 
What happens when the two simple facts intersect? – It becomes strange in a case where global political interests are affected, interests like the implementation of Einstein's standard model of cosmology.
This is my criticism against the article 'The physics of infinity':
1. Reason vs. vanity
Both, Hilbert and Einstein, had their reservations regarding infinity in physics and they seeked to exclude it – each for a different reason. So, what is the difference and what is the point in the current debate?
Hilbert excluded infinity in physics for empirical reasons while Einstein didn't want infinity for formal reasons. Hilberts view was: You can't assign any mathematical attribute to the Universe as a whole when it is probably infinite. Einstein indeed did assign mathematical attributes to the Universe as a whole (even gravity) simply by pronouncing it a bubble (a "spacetime continuum where infinity does not occur"). 
One hundred years later worry is growing that step was pure speculation and vanity.
2. "You are no scientist anymore."
The article culminates in this judgment: Physicists who stay with their axiom the Universe is spatially infinite can't be called scientists anymore. – On the grounds that they can't proof space is infinite.
There are (at least) two things to say about it:
- The components the Universe is made of are principles, laws. The grain and nature of the Universe is to be mathematical, ideational. – And the mathematical idea of space is actually infinite space.
The Universe is in existence because it is a thinkable Universe , because it is a manifestation of sense . I wouldn't say a Universe made of finite space is in no way thinkable – but it must have sense in it. There must be an absolute and essential reason to cut up infinite space (a reason beyond Einstein's vanity). And that is, so far, not the case.
- Is it really true?: A phenomenon of reality that is unprovable, an issue that is unsolvable can't be part of the Universe nor can be an attribute of it? (Because it is proved unsolvable?) . – The article alleges that and it is academical nonsense – virtually not worth to give a response to it.
But the purpose of the article in Nature Physics is not to shine in academic conclusiveness. The purpose is to utter an admonition against physicists who presume a right to stay with the idea of infinity in cosmology.
The authors (and their principals) know the idea of infinity threatens Einstein's cosmology. In the article's text they incorporated some lines that nearly look like equations (to meet scientific expectations) but in fact it is not about science nor logic. It is about politics (the claim to global dominance).
3. "It wasn't me! Hilbert did."
The article hides it's real topic behind a smokescreen of Hilbert's axioms and of "logic". The real concerns are to keep Einstein's "standard model" alive at any cost. The authors of the article impute Hilbert this sentence: "...the Universe must in fact be spatially finite" – they impute it because the authors are too yellow to make themselves their crude political appeal to today's gravitational physicists. – What a dishonest, shady move.
The article reverses Hilbert's view into its opposite. Hilbert says: While it is nowhere observable in the real world infinity is indispensable in theories and in mathematics. The current article tries in reverse to ban infinity in theories (while it becomes virtually observable in the real world, by large telescopes and LHC). 
4. "Einstein is the target for a modern campaign."
No, Einstein is not the victim. Einstein himself carried out the attack on spatial infinity (that was commonly accepted then) one hundred years ago to plant his equations into cosmology. If anything, to ban ideas like infinity from science is the 'modern campaign'.
We live in times of total manipulation and thought control. Institutes like the Max Planck Institute for Gravitational Physics suggest: 'Let's ban some thoughts. It will work.'
Einstein used then the manipulative term of 'clumps' ("galaxies form clusters/clumps – so the Universe as a whole forms a clump").  Today they try this: "What you can't count does not exist" (see the "chain of proof" in the Nature Physics article).
Both cases are crude and untrue. But it works (because it's so crude).
They try to tell us the color yellow does not exist, a state or thinking or science do not exist – the decline of the faculty of reason.
5. Pettiness or infinity
What do you think is the most general defining attribute of the Universe when you think of the trillions of electro-magnetic fields in one single sand grain, when you think of DNA code in the jungle and of galaxies? Is it limitation and pettiness? – No, the one defining attribute of the Universe that is assured is its obscene immoderateness.
The Universe is brought into existence by its conceivableness and its highest law: to form sense. Infinity is downright the essence of conceivableness and sense.
The Universe is made of the categories in which we think of it:
Contractedness or infinity.
The German professor Hermann Nicolai is exemplary for the former.
 Ellis, G., Meissner, K. and Nicolai, H. The Physics of Infinity, Nature Physics, vol 14, August 2018. www.nature.com/articles/s41567-018-0238-1.epdf
 I gave my teaser illustration the title "Three disciples of Einstein at bean counting" (Ellis, Meissner and Nicolai in the hell of infinity). – All rights reserved.
 Hilbert, D. in David Hilbert’s Lectures on the Foundations of Arithmetics and Logic 1917–1933 (eds Ewald, W. & Sieg, W.) 730 (Springer, Heidelberg, 2013). Springer Link
 I focus my response on the German contributor of the article, the Max Planck Institute for Gravitational Physics (the biggest of its kind in the world). The other authors are from Poland and South Africa. Sorry if I miss the main contributor.
 Winter, St. Demiurg Albert E., Die Einsteinsche Kosmologie 1917 - 2017, (in German) 2015-04
 Article about Wheeler, J. Do our questions create the world?, Scientific American, 2018-06
 Winter, St. The Universe as Manifestation of Sense (vixra.org PDF), 2014-04
 Certainly Hilbert made some points that are suited for deriving Einstein's ideology from it. In 1930 Hilbert summarized his understanding of infinity and logic like this (at a speaking engagement in Hamburg):
"Die Physik lehrt, daß ein homogenes Kontinuum, welches die fortgesetzte Teilbarkeit zuließe und somit das Unendliche im Kleinen realisieren würde, in der Wirklichkeit nirgends angetroffen wird. Die unendliche Teilbarkeit eines Kontinuums ist nur eine in Gedanken vorhandene Operation, nur eine Idee, die durch unsere Beobachtungen der Natur und die Erfahrungen der Physik und Chemie widerlegt wird. Andererseits stellen sich in der Astronomie schwerwiegende Bedenken gegen das Vorhandensein des unendlichen Raumes, also der Unendlichkeit im Großen, ein. Auch all unser Handeln ist finit, und das Unendliche findet darin keinen Platz. Das Unendliche findet sich nirgends realisiert; es ist weder in der Natur vorhanden, noch als Grundlage in unserem verstandesmäßigen Denken zulässig. Die bedingungslose Anwendung des Tertium non datur und der Negation können wir aber nicht entbehren, da sonst der lückenlose und einheitliche Aufbau unserer Wissenschaft unmöglich wäre. Das Operieren mit dem Unendlichen muß also durch das Endliche gesichert werden."
 The Universe as a whole can't form a clump, settle from something else, because this 'something else' does not exist. (Empty space isn't this 'something else'. Space is part of the Universe itself.)
Of course the clump topic is Einstein's metaphor for 'gravity of the Universe' (the Friedmann equation). – But there is no gravity of the Universe.