THE INCOMPLETENESS FILE: What the heck is a "formal system?"

THURSDAY, SEPTEMBER 20, 2018

Once again, Joe Average won't know:
Friend, are you a general reader? That is to say, are you a non-specialist in the fields of mathematics, mathematical logic, theoretical physics and the like?

Friend, if you're a general reader, let's consider the title essay of Jim Holt's new book.

The new book is called When Einstein Walked with Godel: Excursions to the Edge of Thought. The title essay is called When Einstein Walked with Godel—and friend, we're telling you this:

Ignore the various things you read about how "readable" Holt's essays are! Friend, if you're a general reader, there is exactly zero chance that you'll emerge from that title essay with even the slightest idea what Kurt Godel's "incompleteness theorems" are alleged to be all about.

As we showed you yesterday, there's zero chance you'll have any idea! Ignore what reviewers have said!

Alas! Several layers of academia, journalism and the publishing world are involved in the creation of this strange state of affairs. Before we look at Rebecca Goldstein's first attempt at explaining Godel's theorems—it's Goldstein's "Godel made easy" book which Holt reviewed in his title essay—let's take a minute to consider, once again, who these high-ranking players are.

We'll start with Godel himself, the man "who has often been called the greatest logician since Aristotle."

Who the Sam Hill was Kurt Godel? As it turns out, he seems to have been mentally ill throughout the whole course of life. (At age 72, he died of self-starvation.)

During his adult years in Princeton, he was famous for believing all sorts of crazy ideas. Among them, perhaps, was his foundational belief in "Platonism"—his ardent belief, in Holt's formulation, that "numbers and circles have a perfect, timeless existence" somewhere. (We're able to access this perfect world through some form of ESP.)

Should it seem strange that our greatest logician can be described in this way? We'll examine that question in more detail next week.

For now, let's continue assembling our list of players. Let's consider the circle of thinkers among whom Godel was moving when he devised his iconic theorems, when he was just 23.

According to the profiles offered by Holt and Goldstein, Godel was moving among the Vienna Circle, a group which is said to have included some of the western world's greatest thinkers. As Europe suffered between two wars, these thinkers were puzzling over how we can know that 2 + 2 = 4. They were also puzzling over how we can know that 4 is an even number.

Later, one of their descendants was puzzling over the question of how we can know that 317 (or 17, or 7) is a prime. Godel, our second greatest logician, was apparently puzzling out these crucial topics too.

Friend, do you find it odd to think that our greatest thinkers were puzzling over such questions? We find that odd (and unimpressive) too, just the way you do!

We find that unimpressive, a point we'll discuss next week. But at this point, we must consider the role in this story which gets played by the publishing industry. We must also consider the work of our own modern-day professors and upper-end journalists.

Our publishing business is awash in "Einstein made easy" books (and the like). None of these books has ever managed to make Einstein easy, including the 1916 "Einstein made easy" book written by Einstein himself.

(Einstein, our greatest theoretical physicist, was not a skilled popular writer.)

No one can understand these books, but professors keep turning them out. They take turns blurbing each other's books, telling us rubes how "lucid" these "accessible" books really are. In response, major reviewers stand in line to say beautifully readable these amazingly easy books are!

As any lover of humor would, we've found this fandango fascinating for a great many years. Next week, we'll consider the real-world problems our savants ignore as they produce unreadable books about 2 + 2 equaling 4 and about how we can know such facts.

Quick question! When our ranking professors behave in these ways, should we really be surprised by the intellectual chaos which characterizes our journalism? When our greatest thinkers behaved (and behave) in these ways, should we really be surprised by the low-IQ mugging and clowning which gets presented on corporate cable each night, as our nation slides into the sea?

(And each morning, on Morning Joe, whose entire panel flipped today concerning the need for an FBI probe of what the accuser has said. The panel moved from yesterday's "no" to today's full-throated "yes." We'd use the accuser's name, except the Times is calling her "Blasey" and the Post is still calling her "Ford.")

When the title essay to his new book first appeared, Holt was reviewing Professor Goldstein's 2005 "Godel made easy" book. Because Goldstein is a highly regarded novelist as well as a ranking philosophy prof, it may have seemed like a great idea to have her write a book about the life and the work of this puzzling, disordered man.

As we noted yesterday, Holt's treatment of the "incompleteness theorems" will be totally incoherent for the general reader. For our money, the general reader won't likely be able to make hide nor hair of Goldstein's treatment either.

Holt wrote a book review for The New Yorker; by way of contrast, Goldstein had written a complete book. In our view, the general reader will have little chance of understanding Godel's theorems from reading that book, but for obvious reasons, we can't reproduce Goldstein's full presentation in the way we could do with Holt.

(We also think the professor went places which we found astounding. "This very sentence is false?" It's stunning to think that ranking professors can still find meaning in places like that. More on that starting tomorrow.)

Where Holt wrote an incoherent essay, Goldstein wrote a hard-to-read book. For our money, the general reader will almost surely emerge from that book with no idea what those "incompleteness theorems" are actually all about.

For today, we'll only show you the way Goldstein introduced the theorems. A person might claim that this is unfair, although we aren't sure it is.

On page 23 of Goldstein's book, she stops discussing Albert Einstein and turns to the young Kurt Godel. As she introduces Godel, she marvels at how young he was when he devised his iconic theorems. She almost seems to say that the theorems are easy to state:
GOLDSTEIN (page 23): He is Kurt Godel, and in 1930, when he was 23, he had produced an extraordinary proof in mathematical logic for something called the incompleteness theorem—actually two logically related incompleteness theorems.

Unlike most mathematical results, Godel’s incompleteness theorems are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof’s overall strategy, delightfully, is not. The two conclusions that emerge at the end of all the formal pyrotechnics are rendered in more or less plain English. The Encyclopedia of Philosophy’s article “Godel’s Theorem” opens with a crisp statement of the two theorems:
Tell the truth! Reading that passage, it sounds like it won't be hard to make Godel easy!

The two conclusions Godel reached "are rendered in more or less plain English," Goldstein writes. "Delightfully," the overall strategy of his proof isn't formidably technical!

Goldstein makes it sound like Godel and his theorems won't be all that hard! Then, she quotes the Encyclopedia of Philosophy's "crisp statement of the two theorems." The passage she quotes goes like this:
GOLDSTEIN (continuing directly): "By Godel's theorem, the following statement is generally meant:

"In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable. (This statement is occasionally referred to as Godel’s first theorem.)

"A corollary to the theorem is that the consistency of a formal system adequate for number theory cannot be proved within the system. (Sometimes it is this corollary that is referred to as Godel’s theorem; it is also referred to as Godel’s second theorem.)"
That quoted passage is attributed the Encyclopedia of Philosophy. We'll suggest you consider this:

According to Goldstein, this account of Godel's theorems has been "rendered in more or less plain English." We trust and believe that you, a general reader, can see that this just isn't so.

How does the Encyclopedia define or describe the first theorem? In plain English, it goes like this:
In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable.
Friend, that passage simply isn't written in plain English. We hope you could already see that.

Citizens, can we talk? The general reader will have no idea what a "formal system" is! Beyond that, this general reader will have little idea what "number theory" is.

In part for these reasons, this general reader won't be able to imagine what a formal system "adequate for" number theory is. The general reader will have no idea what that passage is talking about.

However "crisp" this statement may be, this statement will be clear as mud to the general reader. It contains the kind of technical language which may not look like technical language. But this language is guaranteed to leave the general reader on the outside, haplessly looking in.

Briefly, let's be fair. This passage represents Goldstein's first attempt at describing these iconic theorems. This strikes us as a strange first attempt but, at least in theory, Goldstein could have continued on from there to unpack these theorems in a way the average Joe could actually understand.

For our money, that doesn't happen in Goldstein's book. Along came Holt, to offer the crazily incoherent summary we perused in full in yesterday's report.

On page 26, Goldstein reassures the general reader. She does so in this passage, in which she once again plays the "plain English" card:
GOLDSTEIN (page 26): [Godel’s theorems] are the most prolix theorems in the history of mathematics. Though there is disagreement about precisely how much, and precisely what, they say, there is no doubt that they say an awful lot and that what they say extends beyond mathematics, certainly into metamathematics and perhaps even beyond. In fact, the mathematical nature of the theorems is intimately linked with the fact that the Encyclopedia of Philosophy stated them in (more or less) plain English. The concepts of “formal system,” “undecidable,” and “consistency” might be semi-technical and require explication (which is why the reader should not worry if the succinct statement of the theorems yielded little understanding); but they are metamathematical concepts whose explication (which will eventually come) is not rendered in the language of mathematics.
Finally! Three pages later, Goldstein notes that the general reader has no idea what a "formal system" is. For the record, she offers her first definition of the term on page 129 [sic].

In our view, things don't get a whole lot better for the general reader in what follows from there. Things seem technical all the way down. It seems to us that the general reader will likely be forced to quit.

Citizens, let's review:

Our greatest logician was mentally ill and possessed of crazy ideas. At the heart of his prolix theorems was his apparently crazy belief that numbers and circles live a perfect existence somewhere.

In turn, our philosophy professors seem to have no idea how to explain these prolix theorems (which "say an awful lot") to the general reader. But they produce books which claim to have done that anyway. When they do, journalists rush to say that they understood every word. And it all began with our greatest thinkers pondering 2 + 2.

When we see this cultural pattern unfold, are we surprised by the utter incoherence displayed by lesser thinkers on corporate cable? Are we surprised that our broken, pre-rational public discourse has now helped to place a Donald J. Trump in the White House?

Seeing ourselves from afar, we humans still tend to believe, say and suggest that we're the rational animal. In our view, this profoundly iconic notion qualifies as "Aristotle's [gigantic large howling] error."

Tomorrow, we'll debase Godel a tiny bit more, prepping a bit for next week. We'll also see Professor Goldstein do something amazingly rare.

Tomorrow: A (near) perfect statement by Goldstein

48 comments:

  1. Don't get it? Bring in a consultant. John D Cook quotes Seth Godin:

    https://www.johndcook.com/blog/2013/12/03/heisenberg-godel-and-chomsky-walk-into-a-bar/

    ReplyDelete
    Replies
    1. I love it! Thank you.

      Delete
    2. Good day everybody my name is Mrs Caroline Gilli am here to share with you my life experience on how a great man called Dr Alexzander saved me and my marriage.I have been Married & Barren for for 5 years i had no child. i have never been pregnant i was a subject of laughter from my Friends & neighbors, i almost lost my marriage because of this issue.i was so confused that i did not know what to do until i came across this great Dr online and i contacted him at once i was scared weather it was going to work because i never believed things like this before, so i decided to give it a try and i did all what Dr Alexzander asked of me and today to my greatest surprise i took in the first time and i gave birth to a bouncing baby boy and now my marriage that was about crashing before is now restored. my husband now love and want me better, Am so happy for everything that have been happening in my life since i met this Dr Alexzander.
      I want to tell all the women/men out there who have a similar situation like mine,that the world is not over YET they should dry up their tears and contact this great man and their problem will be gone or are you also having other problems you can also contact Dr Alexzander, here is how you can contact him alexzanderhightemple@gmail.com or contact him via his whatsapp phone number +2348075823891.
      Thank Dr Alexzander for everything you did in my marriage.
      Thanks
      Caroline Grilli

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      Delete
  2. First, it is illegal for universities to discriminate on the basis of disability when hiring professors, and schizophrenia is a disability. If that professor can otherwise do his or her job, having a mental illness is not a legitimate basis for refusing to hire.

    Second, Somerby has indicated his disdain for Kavanaugh's accuser today:

    "We'd use the accuser's name, except the Times is calling her "Blasey" and the Post is still calling her "Ford.")

    Silly woman, to confuse things by having two names. Her professional name, the one she uses in her work and as a professor is Blasey. Her married name is Ford. Women need to keep their professional names in academia because any publications written before they were married need to be accessible to other researchers. If you change your name, any publications appearing under a different name will be lost. Since publications are how you get hired, promoted, and awarded grant funding, they are important and need to be easy to find. Continuity in developing a professional reputation is harder for women who change their names. Hence the confusion that Somerby apparently thinks is insurmountable. Which name you use depends on the context you are talking about someone within. Emily Post understands. Why not Somerby (who blames the newspapers for choosing different names, as if this were a fault or a problem).

    Women fought to use their own names back in the 70s but that effort seems to have been abandoned by many young professionals. The difficulties encountered when navigating bureaucracies are huge when you try to use two names. I believe men have deliberately made this much harder than it needs to be, including men like Somerby who pretends to be confused about who this Kavanaugh-accuser is. Once again, a female professor is being too obscure for Somerby to follow, poor guy.

    ReplyDelete
    Replies
    1. Bitch, shut up and go mop the floor.

      Delete
    2. Better trolls please.

      Delete
    3. @12:58
      Ah, the inevitable representative of the Average Joe contingent makes his appearance.

      Delete
    4. Do some cooking and cleaning woman. Men are smarter than women. Not in the kitchen of course but ... where is the female Mozart, Melville or Mantle? There ain't any so shut up.

      Delete
    5. Do some yard work and change your truck's oil, man. Women are better than men. Not in the garage of course but ... where is the male Mother Theresa, Pavlova or Streep? There ain't any so shut up.

      Delete
    6. @3:26: where is the female Hitler, or Stalin, or Mao, or Pol Pot? Think about that one.

      Delete
    7. Interesting choices:

      1. Melville died in 1891, poor, his books out of print.

      2. Mozart had a talented sister who was not permitted to achieve in music: https://www.theguardian.com/music/2015/sep/08/lost-genius-the-other-mozart-sister-nannerl

      3. Mickey Mantle was a great player but he was also an alcoholic. His restaurant businesses failed and he ultimately supported himself selling sports memorabilia. Kind of sad.

      Lately I have been liking the American Ninja Warrior competition because men and women compete on the same obstacle courses instead of segregating men and women as most sports do. Why should so many activities of society be sex-segregated?

      Delete
    8. This comment has been removed by the author.

      Delete
    9. For 12:58

      https://www.youtube.com/watch?v=mAHfo49iN1w

      Delete
    10. Theodore P Hill studied the 'Greater Male Variability Hypothesis'.

      https://quillette.com/2018/09/07/academic-activists-send-a-published-paper-down-the-memory-hole/

      Delete
    11. They died broke and drunk? Sweetheart, they left footprints in the sands of time! You think Meryl Streep comes anywhere close? Mother Theresa!!?! Oh my God. Your idiocy and lack of history is mind melting!

      Women are made to stay at home and cook and clean. Meryl Streep you put on par with Mozart? Or maybe she's even better than Mozart because of ... she will have more money when she dies and money is what matters, right? Terribly sad the lack of insight. To even begin to compare women with men in the arts in a fools errand. Woman have never and will never hold a candle to men in the arts, humanities, engineering, science - you name it.

      Now bitch, shut your dumb fucking mouth and enjoy the world you live in - the world that allows you to kick back and watch the culture rot American Ninja Warrior (I can only imagine what that is) because the house you watched it in, the car outside, the furniture you sat on, television you watched it on, the industrial food you crammed down your fat ugly face while you watched it, the toilet and plumbing that you used to take a big shit later after it finished where all conceived, built and are maintained entirely by men.

      Delete
    12. This model gives support to a very old debunked theory suggesting male superiority due to greater variability in intelligence. It was addressed in the early 20th century by Leta Hollingsworth. If the study was not spiked for technical reasons, it clearly has an agenda, so activists are not the only ones. Science answers questions -- it doesn't justify propaganda. In other words, you don't manufacture models to prove cherished views, such as male superiority.

      Delete
    13. Be civil or go away. Tired of being called names.

      Delete
    14. It's not suggestiing male superiority, it is suggesting a reckless male insanity that leads to the highest of highs and the lowest of lows. And the Ice Capades.

      I'll try to be civil but you have to stop being so goddamn stupid.

      Delete
    15. God ruined a perfect asshole when she gave you teeth.

      Delete
  3. "In our view, this profoundly iconic notion qualifies as "Aristotle's [gigantic large howling] error."

    No, Aristotle's gigantic error was that women have fewer teeth than men, or that certain people deserve to be slaves, or that women are despicable compared to men. Much bigger errors to choose from than a remark about human rationality that people aren't even sure is attributable to Aristotle.

    ReplyDelete
  4. If famous authors who are book reviewers read every word of the books they write blurbs about, they would have no time to write their own books. Do we believe that Paul Newman tasted every jar of salad dressing that had his name on it? Does Matthew McConaughey drive around the back roads of Texas each night, musing on life? I doubt it, but Somerby seems to think dustjacket blurbs must be gospel, or what's the world coming to?

    ReplyDelete
  5. "In turn, our philosophy professors seem to have no idea how to explain these prolix theorems (which "say an awful lot") to the general reader."

    Is Somerby really complaining, like Emperor Joseph II, that the theorems have too many words?

    ReplyDelete
  6. Let's see; Goldstein's book has 316 pages. Divide by 5, you get 63.2, so that's grist for a little more than 63 weeks of TDH's book review, one page at a time.

    And we already understood, on day one, that Somerby hated the book. And the rest is superfluous.

    (Should I define "superfluous?" That may be hard for the Average Joe to understand, and it might make him angry that I have used an uppity word that he doesn't understand.)

    ReplyDelete
  7. "This very sentence is false?" It's stunning to think that ranking professors can still find meaning in places like that.
    How can someone graduate from Harvard with a degree in Philosophy without being familiar with the Liar Paradox?

    ReplyDelete
    Replies
    1. Did he even graduate?
      Perhaps he made D's in his classes. Or maybe "P" in a pass/fail system.

      Or perhaps he is insulting your intelligence, David, and mine, by putting on an act.

      Delete
    2. https://www.yahoo.com/news/exclusive-immigrant-children-detention-hhs-cuts-funds-programs-like-cancer-research-230259583.html

      Delete
  8. Somerby seems to be solicitous of the Average Joe here. Is Joe in the same category as the Common Man, subsets of which surely contain Real Americans?

    Now, I don't care one way or the other whether Somerby likes Goldstein's book. If he can make the case that it is poorly written, then fine. It's actually not likely that the Average Joe would bother reading it in the first place. After all, Goldstein doesn't call her book "Incompleteness: Gödel made easy" or "Gödel for dummies."

    My problem is with his treatment of Gödel. Somerby seems intent to demean him along with Goldstein. But Gödel is a much more formidable figure than Goldstein; he was a brilliant mathematician whose views deserve better than Somerby's ridicule. From the get-go he has emphasized Gödel's supposed mental illness and has allowed himself to describe Gödel's beliefs, perhaps following Goldstein's lead, in the most childish, ridiculous way. Gödel's thinking cannot be reduced to the cartoonish caricature that Somerby offers here. And Gödel's achievements are profound, and the mathematically illiterate Somerby owes it to his readers, and frankly to himself, to give Gödel a fair hearing, whatever one thinks of Goldstein's book.

    ReplyDelete
    Replies
    1. Yeah... I have to agree with that. Here's a short sample of I found on Wikipedia:

      The first incompleteness theorem shows that the Gödel sentence GF of an appropriate formal theory F is unprovable in F. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (Smoryński 1977 p. 825; also see Franzén 2004 pp. 28–33). For this reason, the sentence GF is often said to be "true but unprovable." (Raatikainen 2015). However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence GF may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as primitive recursive arithmetic, which proves the implication Con(F)→GF, where Con(F) is a canonical sentence asserting the consistency of F (Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403).

      https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems


      Are mathematical equations typically described as “sentences?” Wikipedia is often unreliable. As a member of the illiterati, I’d like to know. If it is true, then language itself is a form of mathematics. Or am I just off the rails here?

      Apparently Gödel’s theorems have had profound implications in computer science (and you may be the anon who pointed this out several posts ago), but it’s completely inaccessible to me as a general reader.

      That seems to be Somerby’s point.

      I am, however, going to continue to parse this thing. Which I’d never even heard about before.

      Leroy

      Delete
    2. Leroy -- I'm not an expert in Godel's Incompleteness, but I think I understand what it means. At one time, I had a decent general understanding of the proof.

      Yes, equations are sentences. When you write, "1 + 2 = 3", 1,2, and 3 are nouns; + is a conjunction; and = is a verb.

      The problem IMHO is that reviewers don't understand Godel Incompleteness, and they don't expect to understand it. So, they have no way to judge how lucid its explanation is.

      Delete
    3. https://www.yahoo.com/news/exclusive-immigrant-children-detention-hhs-cuts-funds-programs-like-cancer-research-230259583.html

      Delete
  9. Why would New Yorker readers care about "lucid" and "accessible" books in the first place? These are not general readers. These are smart guys. Above the rank and file.

    ReplyDelete
  10. So Goldstein says "concepts whose explication (which will eventually come)", then Somerby says "For the record, she offers her first definition of the term on page 129 [sic].", and then fails to quote her definition from page 129, thus preventing Howler readers who have not read Goldstein's book from deciding if she succeeds or not. He just tells us it's a failure. Somerby gets a D- on his book report.

    ReplyDelete
  11. You like self-referential sentences? So do I!

    This sentence is true.
    Self-consistent but vacuous.

    This sentence is false.
    Self-contradictory.

    This is a grammatical English sentence.
    Meaningful and true.

    This is a grammatical Portuguese sentence.
    Meaningful but false.

    This sentence flies me to the moon.
    Grammatical, self-consistent, but absurd.

    This sentence? Ungrammatical!
    Ungrammatical, self-consistent, true.

    This salad of words.
    Ungrammatical, self-consistent, true.

    Now you try.

    ReplyDelete
    Replies
    1. I highly recommend the essays in Godel Escher Bach by Douglas Hofstadter. It includes lots of stuff like this as well as all kinds of other fascinating stuff. I loved the discussion of Chopin's music, for example.

      Delete
    2. https://www.yahoo.com/news/exclusive-immigrant-children-detention-hhs-cuts-funds-programs-like-cancer-research-230259583.html

      Delete
  12. The Average Joe is much more likely to read the next travesty by Ann Coulter or D'Souza than some brainiac book about someone whose name contains two dots over its "o." And those books do far more damage to the brains of the Average Joes and to our discourse than any attempt, failed or otherwise, to respectfully examine the mind and thought of a lonely mathematical genius.

    ReplyDelete
  13. Do not objectify women.
    Has "women" as the object.

    ReplyDelete
    Replies
    1. https://www.yahoo.com/news/exclusive-immigrant-children-detention-hhs-cuts-funds-programs-like-cancer-research-230259583.html

      Delete
  14. Isn't it possible that the reviewers who claimed they liked Goldstein's book really liked it?

    Is it not probable that a publisher will only include positive reviews on the book cover and in marketing materials?

    How many critiques of Goldstein's book were published, and what percentage of them was positive?

    Without knowing the answers to these questions, one cannot honestly infer the existence of some cabal whose goal is to give automatic positive reviews to the authors on a notional list of "approved" authors.

    ReplyDelete
    Replies
    1. Somerby lives in an odd world.

      Delete
    2. So do we: our President is a fat coward.

      Delete
  15. Does everything in this post have to be crap or a lie?

    "We'd use the accuser's name, except the Times is calling her "Blasey" and the Post is still calling her "Ford.""

    That is unadulterated bullshit. Both publications refer to her as her lawyer does as Dr Christine Blasey Ford. Google it.

    ReplyDelete
  16. And then there's Joe Average...what about Jane Average?

    Kevin Drum today shows that men and women don't think identically about some subjects. There are probably others, such as what happened to Blasey Ford. By ramming the appointment through congress, McConnell may be condemning the Republican party to the trash heap. Women care about how they are treated, and they do vote. After putting small kids in internment camps, losing their parents, this may be a final straw for Republican women. Republicans should be treading much more carefully on this issue, in my opinion.

    ReplyDelete
  17. Good day everybody my name is Mrs Caroline Gilli am here to share with you my life experience on how a great man called Dr Alexzander saved me and my marriage.I have been Married & Barren for for 5 years i had no child. i have never been pregnant i was a subject of laughter from my Friends & neighbors, i almost lost my marriage because of this issue.i was so confused that i did not know what to do until i came across this great Dr online and i contacted him at once i was scared weather it was going to work because i never believed things like this before, so i decided to give it a try and i did all what Dr Alexzander asked of me and today to my greatest surprise i took in the first time and i gave birth to a bouncing baby boy and now my marriage that was about crashing before is now restored. my husband now love and want me better, Am so happy for everything that have been happening in my life since i met this Dr Alexzander.
    I want to tell all the women/men out there who have a similar situation like mine,that the world is not over YET they should dry up their tears and contact this great man and their problem will be gone or are you also having other problems you can also contact Dr Alexzander, here is how you can contact him alexzanderhightemple@gmail.com or contact him via his whatsapp phone number +2348075823891.
    Thank Dr Alexzander for everything you did in my marriage.
    Thanks
    Caroline Grilli

    ReplyDelete

  18. I'm Olivia Megan from United State,I'm happy that my husband is back into my life after 2 years of divorce, Dr.AKHERE brought my husband back today and i am so excited. I got DR AKHERE email online when a lady was testifying about the strong spell caster who restored her marriage then I said to myself since he helped her, he can also help me,so i emailed him and told him the pain that I was going through,and he told me what to do and i did it,Then he did an urgent Love spell for me. 48 hours later, my husband came back home and with lots of love and joy,and he apologized for his mistake,and for the pain he caused me. Then from that day,our marriage was now stronger than how it were before, All thanks to DR AKHERE. Our family is complete again. If you are going through Divorce/Broke-up since DR AKHERE helped me, he can also help you..email him at: AKHERETEMPLE@gmail.com ,Thank you DR AKHERE for saving my broken Marriage and brought my husband back to me.
    Email him: AKHERETEMPLE@gmail.com
    or
    call/whatsapp:+2349057261346

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  19. Hello,

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